View
6
Download
0
Category
Preview:
Citation preview
CIVil ENGINEERING STUDIES .pt; 'l'. ;--- :;-
0'~-~~~ :---:~.; STRUCTURAL RESEARCH SERIES NO. 129 Y
SUMMARY OF INVESTIGATIONS IN NUMERICAL
AND APPROXIMATE METHODS OF STRESS ANALYSiS
I By
W. J. AUSTIN, T. Y. CHEN,
and
A. S. VELETSOS
Approved by
N. M. NEWMARK
FINAL REPORT
to
OFFiCE OF NAVAL RESEARCH
Contract N6ori-071 (06), Task Order Vi
Project NR-064-1 83
UNIVERSITY OF ILLINOIS
URBANA, ILliNOIS
I.
II.
TitBLE OF CONTENTS
INTRODUCTION. < 0
llNESTlGATIONS IN srRESS A..T'ffiLYSIS .
1. Plane stress . 0 0
2< Radially Symmetric Stress Problems
3 " Genera.l ized Plane Stre s s .
4< Torsion. 0 •
5 • Curved Beams 0
6.
7·
8.
9·
Beams on Elastic Supports ...
Flexure of Plates .. 0 • 0
(a) Analysis of Plates Continuous Over Flexible Beams. 0 • • 0 • • • • •
(b) Skew Slab and Girder Floor Systems. .
Stress Analysis of Stiffened Shell Structures ..
Analysis of Shells . 0 0 • •
(a) Pressure Vessel Heads . (b) (c) (d) (e)
Domes Under Symmetrical Loading . Cylindrical Shell Roofs . . 0 • •
Hipped Plates . . 0 0 0 0 • • • 0
Stresses Caused by Initial Irregularities .
III. INVEsrlGATIONS:m BUCKLING A.,~D VIB.RATIONS • . • • 0 •
10. Numerical Procedures for Vibration Problems ...
11. Natural Frequencies of Elastically Restrained Bars 0 • " 0 • ., 0 • • • • • 0 0
12. Vibration of' Continuous Flexural Systems ..
13. Effects of Rotatory Inertia and Shearing Distortion on Vibration of Bars.. . ....
14. Vibration of an Rlastic Solid ..
15· Buckling of Elastically Restrained, Non-Prismatic Bars 0 • C <> 0 0 0 • 0 0 • 0 <>
t I I
I ..
Page
1
5
5
9
13
14
15
16
20
20 23
25
26 26 27 28 29 30
32
32
34 .. ..I ..I
36 .. :s ) ".
II
38 , ~
39 ... :-
40
TABLE OF CONTEIfilS (CONT INUED )
16. Buckling of Plates .....
Page
41
17. Torsional-Flexural Buckling of Beams and Coltun11s . . . . . . . • . . . . . .. . . . 42 (a) Development of Numerical Procedures. . 43 (b) Elastically End-Restrained I-Beams 44 (c) Beams Supported by Cables. . • . . . . .• 45
rI. RESPONSE OF srRUCTURES UNDER TRANSIENT LOADING. 47
v.
18.
19·
20.
21.
Development of Numerical Procedures of Analys is .. . . . • . . . . (a) Procedure Using Taylor 1 s Expansion (b) Newmarkfs ~ Method ..... . (c) Extension of t3 Method - Use of Dynamic
Load Fac tor. . . . . • . . . . .. . . • • •
Review of Numerical Procedures of Analysis.
Response of structures to Earthquake MOtions. .. (a) Influence of Ductility on the Response
of structures to Earthquake Motions. . . . (b) Aseismic Design of Elastic structures
Founded on Firm Ground . . . 0 • • .. • • •
(c) Response of a Typical Tall Building to Actual Earthquakes . . . . . 0 • • • • • •
(d) Distribution of Extreme Shear in a Tall Building Subjected to Earthquakes.
Response of structures to Blast Loading . . (a) Response of Simple structures to
D,ynamic Loads .......... .. (b) An Engineering Approach to Blast
Resistant Design . . . • . • . . ( c) Blast Loading Transmitted From Walls to
Building Frames. . .. . • • • • . •• (d) Influence of Foundation Coupling on the
D,ynamic Response of Simple Structures. . • (e) Effect of Vertical Loads on the D,ynamic
Response of structures .
MISCELLANEOUS
22. Bounds and Convergence of Relaxation and Iteration Procedures ......... 0 •
47 4B 49
50
51
53
53
55
56
57
58
59
61
62
63
64
66
66
TABLE OF CONTENTS (CONTnruED)
23. A Method of Numerical Integration for Transient Problems of Heat Conduction. . • . . . . . . . . 67
24. studies on the Use of High Speed Digital Computing Machines • • . . . . . . . . •
LIsr OF TABLES
I. List of Theses Prepared Under Sponsorship of Contract
68
N6ori-7l, Task Order VI . ...... 72
II. List of Technical Reports Issued Under Contract N6ori-7l, Task Order VI . . • . . • . 75
III. List of Publications Prepared Under Sponsorship of Contract N6ori-7l, Task Order VI . . . . . . . 78
CHAPI'ER I
INTRODUCTION
This report is in fulfillment of the requirements of Contract
N6ori-07l(06), Task Order VI, Project Designation Numbers NR-035-l83 and
NR-064-l83, between the Mechanics Branch of the Office of Naval Research
and the University of Illinois. The purpose of this contract was to con
duct research in the field of numerical and approximate methods of analy
sis of structural and machine elements and in related fields of mechanics.
This research program included studies of the range of appli
cability and the relative merits of available numerical and approximate
methods of stress analysis, the development of ne\{ and more efficient
numerical methods, the preparation of programs for use on the ILLIAC,
the high speed digital computer of the University of Illinois, the solu
tion of important practical problems o.f stress analysis, instability,
vibration, impact, etc., and the publication of the results of these
studies. In general, two different types of numerical procedures have
been considered. The first concerns procedures which are rapid and rea
sonably accurate for preliminary design. Attention was devoted in this
aspect of the work to procedures which do not require complicated com
puting equipment. These procedures are capable of application with only
a slide rule or at most an ordinary desk computing machine. The second
type of procedure that has been investigated is intended primarily for
research, to give solutions o.f any desired accuracy for problems for
which classical methods of analysis are difficult to apply. This type
of procedure is especially well suited for use with high speed electronic
computing equipment.
2
The contract for this program was initiated on June 1, 1946 and
terminated on November 30, 1955. Since the termination date research on
numerical and approximate methods of analysis has been continued under
Contract Nonr-1834(03), Task Order 3, Project Designation NR-064-183.
Quarterly status reports have been issued regularly since Decem
ber 1948. These reports contain a brief description of the progress of
the work in the various phases of the project, plans for future investi
gations, and relevant administrative details such as changes in personnel.
The investigations of this program were carried on by men work
ing for their M.S. and Ph.D. degrees. A list of thirty-five theses pre
pared under the sponsorship of this proj ect is given in Table I. Of
these) twenty-six were for the Ph.D. degree and nine were for the M.S.
degree. In addition to the students whose names appear in Table I,· there
have been a number of graduate students who worked for a time on Task VI
but are not listed in this report because they were transferred to other
projects where they obtained their degrees. Also there have been a num
ber of undergraduate students who worked as assistants but whose names
are not included herein.
Technical reports have been issued when a specific phase of the
work was completed or when important results meriting wide distribution
were obtained. Twenty-six technical reports have been issued on this
project. A listing of these technical reports is given in Table II. Also
included in Table n are two reports which are of the nature of technical
reports but were issued before the system of technical reports was begun.
3
Twenty-one formal publications, including nineteen papers and
two discussions, have resulted to date from the investigations of this
program. The titles, references to the journal, etc., are given in
Table iII.
The studies in this program have been conducted under the
supervision of ~. N. M. Newmark with the assistance of Drs. W. J.
Austin, L. E. Goodman, T. P. Tung, and A. S. Veletsos.
The writers appreciate greatly the enlightened attitude of
the sponsors in giving complete academic freedom in this study of fun
damental problems in numerical methods applied to engineering. The
studies have not been dictated by external demands nor restricted by
time limitations. It has been possible to make use of many graduate
students thereby increasing greatly their advanced educational training
and at the same time enabling the students to earn part of their expenses.
The aim of the writers to strengthen the fundamental scientific research
program in structural engineering has been made possible by the support
of the Office of Naval Research. The influence of this support has been
felt on the whole field of structural mechanics.
The purpose of this report is to summarize the major investiga
tions conducted under this research contract. This work is presented in
the following four chapters. Chapter n is concerned with investigations
in stress analysis) Chapter III with stUdies of buckling and vibration,
Chapter IV with studies of the response of structures under transient
loading, and Chapter V with miscellaneous investigations of a general
matheIIlatical nature. A short S'l..UllIIl8.ry is given of each investigation
4
including the object of' the study, the methods employed, the nature of
the results, and any important conclusions.
CHAPrER II
INVESTIGATIONS IN SI'RESS ANALYSIS
5
In this chapter are described the investigations conducted in
the field of stress analysis. Included are investigations in numerical
analysis of problems of plane stress, radially symmetric stress, gener
alized plane stress, torsion, curved beams, beams on elastic supports,
plates, shells, and domes. Much stu~ has been expended in this .field
and this effort has resulted in sixteen advanced degree theses.
1. Plane stre ss
As is well known, the solution o.f any plane stress problem must
satisfy the equations of equilibrium and compatibility for all interior
points of the structure considered and also the specified conditions at
the boundaries. Available methods of analysis may be classified from a
physical point of view as follows:
(1) Equilibrium procedures, in which the conditions of equili
brium are expressed in terms of unknowns which automati
cally satisf'y the compatibility requirements, and
(2) Compatibility procedures, in which the conditions of com
patibility are expressed in terms of unknowns which auto
matically satisf'y the conditions of equilibrium.
In either case, the values of the unknowns along the boundary are chosen
so as to comply with the specified boundary conditions.
6
In problems of plane stress, the choice of displacements to
express the equations of equilibrium and the boundary conditions is an
example of the use of the equilibrium procedure. ArJy set of displace
ments automatically satisfies the compatibility requirements. An illus
tration of the application of the compatibility procedure to the same
class of problems is provided by the use of the Air,y stress function.
In this case the biharmonic equation expresses the conditions of com
patibility in terms of a stress function which automatically satisfies
the equations of equilibrium.
Numerical solutions may be made in both the equilibrium and
the compatibility procedures. Numerical methods fall into two general
categories:
(1) Framework analogy methods or physical analogies in which
the solid material is replaced by a system of bars, usu
ally pin-connected, the proportions of which are chosen
so as to represent the physical behavior of the solid as
closely as possible;
(2) Finite difference methods in which approximate solutions
to the appropriate differential equations are obtained.
Four numerical procedures of the framework analogy type were
studied. Two frameworks were considered, a hexagonal framework and a
square lattice network. The hexagonal framework consists of pin-connected
bars formed into equilateral triangles. The six triangular elements baving
vertices at a joint form a hexagon, which represents the basic unit con
sidered in the analysis. The square lattice framework consists of pin
connected bars formed into squares with crossing diagonal bars connecting
7
opposite corners. Both equilibrium and compatibility procedures were
studied for the solution of each type of framework. In the equilibrium
procedure the two equations of equilibrium for each joint are expressed
in terms of the displacement of that joint and the displacements of the
adjacent joints. The two components of displacement at each jOint are
found by the solution of a set of linear algebraic equations. In the
compatibility procedure a set of statically balanced forces are assumed
in the bars of the framework. These stresses must, of course, be con
sistent with the boundary stresses and any internal body forces. In
general these assumed stresses give elongations of the bars which do not
satisfy continuity requirements; that is, the bars cannot fit together
with such deformations. By introducing statically self-balanced sets of
stresses in the bars the continuity requirements can be satisfied with
out disturbing the equilibrium of the joints. Standard correction pat
terns are worked out for each framework and the required values may be
found by the solution of simultaneous algebraic equations.
This study was conducted by E. L. Dauphin and reported in his
If. S. thesis (Item II of Table I). Both framework analogies were found
to be practical methods for plane stress analysis. The compatibility
procedure was found to possess the following advantages over the equili
brium procedure for the solution of the bar stresses in the frameworks.
The compatibility equations are simpler and less changeable near bound
aries than are the equilibrium equations. This feature is very helpful
if relaxation or iterative procedures of solution are used. Another
great advantage of the compatibility procedures is that they result in
8
fewer simultaneous equations. For hexagonal networks less tha...l'l one half
as many simultaneous equations are needed for a compatibility solution
as are needed for an equilibrium solution, since the compatibility solu
tion requires only one equation for each interior jOint and no equations
.for boundary joints. Body forces do not affect the compatibility equa
tions but add an extra term in the equilibrium equations. In the com
patibility procedure, when the solutions of the simultaneous equations
are not exact, as is often the case for iteration or relaxation solutions,
one ends up with a statically balanced set of bar stresses with small dis
crepancies in continuity. This result seems preferable to ending up with
a set of statically unbalanced stresses, which is the case with the equi
librium procedure. On the other hand, the compatibility procedures have
the following two disadvantages: (1) they cannot be used when displace
ments are specified at the boundary and (2) the physical significance of
the correction quantities in the compatibility computations is not as
apparent as that in the equilibrium procedure.
Further studies of numerical procedures for plane stress were
conducted by Z. K. Lee and reported in his M.S. thesis (Item 18 of Table
I). In this investigation the use of finite differences in solving for
the Airy stress function was studied and this procedure was compared with
the square lattice framework method. Several aspects of the Airy func
tion problem were considered. In addition to the derivation of a basic
procedure based on first order differences, a structural analogy consist
ing of a system of bars was devised which leads to identically the same
equations as the finite difference approximation to Airy's function. This
9
analogy is helpful in understanding the nature of the approximations
involved in the finite difference procedure. In addition, the use of
higher order differences was investigated for a few cases and was found
to result in a considerable increase in accuracy. Finally, as a special
case, a procedure for plates with holes was derived. An important aspect
of this investigation was concerned with a comparison between the Airy
function procedure and the square lattice framework method. It was
found that for the same size network both methods give about the same
accuracy. However, the framework analogy method leads to about twice
as many equations as the Airy stress function method. Mr. Lee concludes
that between these two methods the choice is likely to be in favor of
the Airy function method in most cases.
The fundamental ideas of these studies were discussed by Dr.
N. M. Newmark in a paper entitled, "Numerical Methods of Analysis of
Bars, Plates, and Elastic Bodiesu presented at a symposium held at illi
nois Institute of Technology and published in a book entitled, "Numeri
cal Methods of Analysis in Engineering" (Item 2 of Table ill).
2. Radially SymmetriC stress Problems
The object of this study was to develop satisfactory approxi
mate numerical methods for the solution of radially symmetric stress
problems. Two general methods were considered. One is the lattice anal
ogy method in which it is imagined that the solid body can be replaced
by a framework of bars. The second scheme involves the well known pro
cedure of expressing a differential equation in its finite difference
10
form. In both methods the displacements at a discrete number of points
are considered, and a solution is obtained by a process of iteration.
In the lattice analogy method it is assumed that the behavior
of a solid radially symmetric body under a radially symmetric load can
be approximated by that of a framework of bars subjected to the same
load. The area of each bar is determined from a consideration of the
deformation of an element of the solid body and the corresponding ele
ment of the framework, when both are subjected to certain simple force
systems. Equations of equilibrium are written at each joint in terms
of the displacements at the joints. Therefore, two equations are
obtained at each nodal point, and the analysis is made by the solution
of a set of linear, algebraic equations. From the components of dis
placements the forces in the bars of the framework are computed. From
the forces in the bars of the framework one can obtain an approximation
to the actual stress distribution in the solid body.
The lattice analogy method was found suitable for the solution
of mixed boundary value problems. However J the method appears to be
limited, due to time consuming computational work, to bodies whose geo
metric shape is formed of straight radial and axial lines. Curved sur
faces in the framework are represented by a series of steps which follow
in an approxiInate manner the outline of the boundary. Problems solved
with such irre~_~~ boundaries in the analogy method showed high stress
concentrations at the re-entrant corners. The solutions in these cases
were not representative of the actual curved boundary problem. Solutions
with finer subdivisions in the region of the curved surface were attempted,
11
but the results did not appear to justify the lengthy computational work
involved.
In an effort to obtain a more suitable method of defining a
curved surface by a discrete number of points and of simplifying the
solution of curved boundary problems to a point where a good approxima
tion could be obtained with a coarse net in a short period of time, the
lattice analogy method was replaced by the method of finite differences.
In the finite difference method the differential e~uations of
equilibrium in terms of displacements were approximated by finite dif
ferences. Several coordinate systems were used. The use of cylindrical
coordinates gave good results for rectangular boundaries. However,
curved boundary surfaces cannot be suitably treated with a system of
cylindrical coordinates because it is difficult to define the boundary
accurately with a discrete number of network points. An attempt was
made to find the stresses around a notch, but accurate answers could not
be obtained at the root of the notch where the stress gradient is steep.
To obtain a better solution for the notched bar problem an
orthogonal curvilinear system of coordinates was adopted. Such a system
bas certain inherent properties which make it desirable for the solution
of curved boundary problems. No difficulties arise in the delineation
of a curved s~ace by a discrete number of points or in the determina
tion of the finite difference expressions for the boundary conditions in
terms of stress that apply to the curved surface.
The particular coordinate system used was the ellipsoidal. The
network of intersecting traces of byperboloids and ellipsoids on an axial
12
plane yielded a pattern which had the desirable feature of the finer and
denser subdivisions occurring in the region of high stress concentration
at the root of the notch. Good results were obtained with the curvilinear
coordinate solution.
Finally, a method was developed for solving inelastic radially
symmetric stress problems. The yield and flow condition used was assumed
to be a ~ction only of the second invariant of the stress deviation.
To define the start of plastic yielding or to define the flow condition
for perfectly plastic materials the Huber-Mises-Hencky condition was
used. It was assumed that the material was under a state of gradually
increasing load, and the stress analysis was made for a particular stage
in the loading process. It was also assumed that~ (a) the axes of prin
cipal stress and strain coincide, (b) the deviations of stress and strain
are proportional, and (c) the intensity of stress is a completely deter
mined function of the intensity of strain for all complex states of stress.
The basic equations that arise from these fundamental relations and the
equations of statics are linear in form when expressed in terms of dis
placement components. This .characteristic of linearity of the differ
ential equations is a desirable feature as it permits one to obtain a
solution of an inelastic problem by solving a sequence of elastic problems
through a process of successive corrections.
This work is reported in a Ph.D. dissertation by Elio DI Appolonia
(Item 10 of Table I). Part of this work was published in an article by
D'Appolonia and Newmark, (Item 7 of Table III) and this paper was reprinted
and issued in combination with two other papers published at the same time
13
in a technical report by Newmark, D'Appolonia, and Austin (Item 12 of
Table II).
3. Generalized Plane Stress
The two dimensional solution of plane problems in the theory
of elasticity is a mathematical idealization. It only gives reasonable
stresses either in a plate which is very thin or in the neighborhood of
the mid-plane of a very thick plate. For a plate with moderate thickness
neither the plane stress nor plain strain solution furnishes adequate
results.
In an investigation conducted under this program, a numerical
procedure was developed for the three dimensional analysis of a plate.
The method of finite differences was used to obtain a correction to the
existing two d~ensional solution, such that the complete solution sat
isfied all the equations of equilibrium and compatibility as well as the
prescribed boundary conditions. The solution was made using displace
ments as unknowns.
The numerical procedure was applied to the analysis of an infi
nite plate with a circular hole under uniaxial tension, where the dia
meter of the hole is equal to the thickness of the plate. Approximate
solutions to this problem had been published by Green and by Sadowsky
and Sternberg, and these solutions were compared with the numerical results
of this study. The problem was solved numerically with a three dimensional
rectangular network, applicable to any shape of hole, and also with a two
dimensional rectangular network applicable only to this problem in which
14
the variation of displacements, stresses, etc., about the hole is known a
Different size networks and higher order differences were used and a
study of the accuracy of the various possibilities was made.
Although this study was restricted to an elastic, isotropic
material, it appears that it can be readily generalized to account for
conditions such as inelastic deformation and non-isotropy.
This investigation was conducted by T. P. Tung and reported
in his Ph.D. dissertation (Item 29 of Table I). ~. Tung concludes that
the method of finite differences is an adequate method for solving prob
lems which otherwise would be very tedious by classical methods. Rea
sonable results may be obtained without introducing very fine networks.
It is suggested that in general improved results might be obtained if
the derivatives in the boundary equations are approximated by finite
differences including higher order terms than are needed for the equa
tions of the interior points.
4. Torsion
The purpose of this investigation was to develop an approximate
numerical procedure for the solution of Laplace 1 s equation and Poisson's
equation, with special emphasis on the stress function equation for tor
sion. The procedure which was developed depends upon replacing the dif
ferential equation with an equivalent difference equation. The equation
is expressed in terms of the differences between ordinates to a surface
rather than in terms of the ordinates themselves, which is the usual pro
cedure. For any particular problem this procedure results essentially
15
in a set of simultaneous, linear, algebraic equations. The equations are
never actually written but the solution is made on a s~ple diagram by
a process of iteration or relaxation. The solution is presented as a
flow analogy in a network o.f pipes. p...:n important aspect of this inves
tigation was the development of a convenient procedure for the problem
of torsion of a bar of constant cross-section having one or more internal
longitudinal holes.
Throughout this study an effort \vas made to present the proce
dures in engineering terms so that they would be readily understood and
used in practice.
The findings of this investigation were presented in an M.S.
thesis by Eo/ C. Colin (Item 9 of Table I), in two technical reports by
Colin and New.mark (Items 27 and 28 of Table II), and in a published paper
by Colin and Newmark (Item l of Table III). The procedures developed in
this study are also described in a general paper on numerical methods of
stress analysis by N. M. Newmark (Item 2 of Table III).
5 . Curved Beams
This study vres concerned with three aspects of the stress anal
ysis of curved beams. First, a theory of flexure of curved beams was
developed, by which the normal stresses in curved beams due to direct
thrust and due to bending about two axes, one radial and the other per
pendicular to the plane of curvature, may be readily calculated 0 This
method employs the concept of the transformed section, as originally sug
gested by Rardy Cross. The method is based upon a generalization of the
16
assumptions of the Winkler-Bach theory for curved beams. With this pro
cedure the computations are almost identical to the corresponding ones
for straight beams. stresses in curved beams due to shears and twisting
moments were not considered in this investigation.
A second aspect of the study was the determination of the shear
center for unsymmetrical thin-walled open cross-sections of curved beams.
Formulas were derived for the shear flow in a thin-walled open cross
section and for the location of the shear center for certain types of
channel sections and other cross-sections.
Finally, the transformed section method was applied to the prob
lem of' secondary bending of flanges of thin-walled symmetrical I and T
sections. A different method for dealing with this problem had been pr.e
sented previously by H. H. Bleich.
This study was made by H. K. Yuan and reported in his Ph.D.
dissertation (Item 35 of Table I).
6. Beams on Elastic Supports
Two investigations were conducted on the development of numeri
cal methods for the analysis of beams on elastic supports. The first
study was concerned '\vith beams resting on an elastic foundation of the
Winkler type, whereas the second study was concerned with beams resting
on an elastic continuum.
In a foundation of the Winkler type, the reaction provided by
the foundation on the beam at any point is proportional to the deflection
of the beam at that point. Two different numerical procedures were inves
tigated for the analysis of beams resting on such a foundation. The most
17
versatile procedure was found to be a combinat~on of the Stodola succes
sive approximations method with Newmarkqs numerical integration method
for ~inding deflections of beamsft With this procedure one may consider
in a straightforward manner any desired variations in loading, stiff'ness
of beam, and modulus of foundation. Also one may conveniently take into
account the effect of spring reactions from supporting members, and of
non-linear or plastic behavior of the beam and/or supporting medium.
Better accuracy is obtained with this procedure than can be obtained by
most other numerical methods, for the same number of divisions in the
length of the beam.
The success of this method depends entirely on the rate of con
vergence of the successive approximations to the desired solution. When
rapid convergence is attained the method is superior to other numerical
procedures. On the other band, when the successive approximations diverge
ba~ a solution can be obtained by this method only with great difficulty.
Because of the importance of this feature, much time was spent in a study
of convergence and in attempts to devise schemes to force convergence for
naturally divergent problems. It was found that the rate of convergence
or divergence depends upon a single dimensionless parameter, which is the
ratio of the foundation modulus to the stiffness of the beam. The range
of this parameter for which the successive cycles converge has been clearly
delineated and, based upon the theoretical results, a method. vl8.S developed
for obtaining a good initial approximation.
An alternate method o.f analysis was developed for problems which
are so divergent that the successive approximations procedure is not fea
sible. This procedure makes use of a step-by-step solution of the equations
18
in the manner analogous to that of an initial value problem. This pro
cedure does not involve difficulties of convergence, but is restricted
to linear problems.
This investigation was reported in an M.S. thesis by F. W.
Schutz (Item 25 of Table I).
In the second study a numerical procedure ,{as developed for
the analysis of beams resting on a semi-infinite elastic-solid medium.
For this problem the force exerted by the supporting material at any
point depends not only on the deflection at that point but also on the
deflection at every other point. When the subgrade is treated as a
true elastic continuum characterized by the modulus of elasticity and
Poisson's ratio, the problem is governed by an integro-differential
equation of which very few exact solutions are known.
In this investigation the beam is treated as an elastic strip
following the elementary beam theory. The subgrade is considered either
as a plane plate or as a three-dimensional elastic body. Two procedures
were developed. The first approach uses the method of successive approx
imations in conjunction with Newmarkis numerical integration procedure.
This procedure is very similar to that described by Schutz in the first
investigation, except for the determination of the sub grade reactions.
An alternate procedure utilizing a solution of simultaneous equations is
proposed for beams of uniform section, and especially for symmetrically
loaded beams resting on a very stiff subgrade 0 In this alternate method
influence coefficients are used.
19
The key to these methods is the determination of the subgrade
reaction induced by a given set of deflectionso This is found by approx
imating the actual subgrade reaction by block loadingso The deflection
at each of a series of node points along the surface of the semi-infinite
elastic solid due to block loads at those points is found by the theory
of elasticity. These deflections form a matrix which may be inverted
to give the forces on the sub grade required to produce the given set of
deflections. This method was found to be accurate enough for practical
purposes.
The method developed is quite general. It has been applied to
the problem of buckling of beams on elastic -solid sub grades, and can easily
be extended for solving problems of rectangular plates supported on an
elastic solid. In the latter problem the ordinary plate theory is assumed
to be valid and the subgrade is treated as a three dimensional elastic
body. The plate is divided into a convenient network, with the node point
considered to be the center of each element. The calculus of finite dif
ferences may be used to express the differential equation of the plateo
The sub grade reaction distribution is approximated by block loads over
each element. The deflection at the node point due to the block loads
on the elements is found by the theory of elasticity 0 These deflections
form a matrix which is inverted to give reactions in terms of deflections.
The remaining steps are the same as those for the beam-subgrade problem.
This investigation was conducted by D. H. Lee and reported in
his Ph.D. dissertation (Item 17 of Table I).
20
7. Flexure of Plates
Two investigations were conducted to determine the state of
stress and deflection in medium-thick elastic plates supported by flexi
ble beams. The specific problems considered are discussed in the follow
ing sections.
(a) Analysis of Plates Continuous Over Flexible Beams
This study was concerned with the analysis of an interior panel
of a plate continuous over a rectangular grid of flexible beams which are
supported at their intersections by columns. It is considered that the
plate is uniformly loaded over its entire area, that it has a large num
ber of panels in both directions, and that parallel beams are of equal
stiffness and uniform spacing. Under these conditions, the distribution
of moments in all interior panels may be assumed to be identical and only
one interior panel need be considered. In this investigation the ordi
nary theory of medium thick plates was used. In addition, it was assumed
that the widths of the beams and the cross-sectional dimensions of the
columns are small compared with the panel dimensions. While this latter
simplification has little effect on the deflections and moments near the
center of the panel, it does, in some cases, lead to excessively large
bending moments near the columns.
The system was analyzed by means of the Rayleigh-Ritz energy
procedure, us ing a set of polynomial functions due to W. J. Duncan. The
second derivatives of these functions are normalized Legendre polynomials,
and the first function of the set is proportional to the deflection of a
21
uniformly loaded fixed-ended beam. The properties of these functions were
found to be very appropriate for the present problems.
Numerical calculations have been made for panels with side ratios
equal to 0.5, 0.8, and 1.0" and various ratios of beam rigiditiesc For
the square panels, the value of the ratio of rigidities was varied from
zero to infinity. For the rectangular panels, it was assumed either that
all beams are of equal rigidity or that all beams have equal moments of
inertia. For each structure considered, numerical values were computed
for deflections at three locations (midpoint of the panel, and midpoint
of the beam in each direction), for longitudinal and transverse moments
in the plate at four locations (midpoint of the panel, corner of the panel,
and midpoint of the beam in each direction), for the average moments in
the column and the middle strips at the center lines of the panel and over
the beams in each direction, and for the moments in the beams at midspan
and at the columns. The values given are .for Poisson i s ratio equal to
zero; however, they can be easily converted to any desired value of Poi
sson t s ratio. The numerical data obtained have been presented in both
tabular and graphical forms.
In general, the results presented are those obtained using nine
terms of the series for the deflection function 0 Thus each case involves
the solution of a set of nine simultaneous equations. However, in order
to test the accuracy of the results obtained, two additional solutions
were obtained for each case considered in the present work, using only
one and four terms, respectively, in the approximating series of the
deflection function: By inspecting the rate of convergence of a sequence
22
of such solutions, some indication of the accuracy of the results may be
obtained. It has been found that the convergence of the sequence of
solutions is very good for the clamped plate, and it becomes even better
as the ratio of beam rigidity to plate rigidity approaches unity 0 As
this ratio is decreased, the convergence becomes poorer. The solutions
show the poorest convergence for the plate supported on columns only 0
Also the convergence is better for square or nearly square panels than
for long and narrow panels.
Comparisons were also made between the energy solutions and
known analytic solutions for the two limiting cases of a plate clamped
against both rotation and deflection along its edges, and a plate sup
ported by a rectangular array of' columns without connecting beamso The
agreement of the results has been found to be f'airly good Q Comparisons
were also made with solutions obtained by means of finite differences.
It was found that the energy solutions obtained with only four terms in
the series (four simultaneous equations) are generally more accurate
than the finite difference solutions obtained with 24 simultaneous equa
tions.
The solutions obtained may be applied in the analys is of prac
tical structures such as two-way concrete floor slabs, provided that
proper account is taken of the dissimilarity between the actual condi-
tions and the idealized conditions asswlled in the present study 0 These
solutions, by appropriate superpositions with other known solutions, may
be used also to find the influence of loads which are not uniformly dis
tributed over the entire area of' the floor slab.
23
This study has been described in a Ph.D. dissertation by J. G.
Sutherland (Item 27 of Table I) and in a technical report by Sutherland,
Goodman, and Newmark (Item II of Table II). An exact solution found in
this study was published by Sutherland (Item 9 o.f Table nl).
(b) Skew Slab-aDd-Girder Floor Systems
This investigation was concerned with the behavior of simple
span skew slabs supported on flexible girders. The girders were consid
ered to be identical, uniformly spaced, parallel to one pair of sides
of the slab, and simply supported at their ends. No analytical solu
tions for this type of structure had been available. In this study, the
numerical method of finite differences was used.
Difference equations were derived for a general system of skew
coordinates to permit the analyses of the structures for any angle of
skew, ~, ratio of girder spacing to span, bfa, and relative stiffness
of the girders and slab, H. The equations derived are applicable to
structures having any number of girders.
As an application of the equations to a practical problem, a
special type of structure often used in highway bridge construction, the
skew I-beam bridge, was analyzed. A total of 18 skew bridges, each having
five girders, were considered. The physical characteristics of the bridges
were defined by combinations of the following variables~ cp = 30, 45, and
60 deg.; H = 2, 5 and 10 for b/a = 0.1; and H = 1, 2, and 5 for b/a = 0.20
With the use of difference equations, influence values for moments in the
slab and girders at various locations produced by a moving unit concen
trated load were determined. The solution of simultaneous equations \vas
24
carried out by the ILLIAC. Numerical data were presented in tabular as
well as graphical form.
Maxtmum live load moments at mid-span of girders produced by
standard highway truck loadings were then computed for 72 structures
having beam spacings of 5, 6, 7, and 8 ft. for different values of bfa,
H, and~. Based on these results, empirical relationships were devel
oped for estimating the maximum live load moments at mid-span of any
skew bridge having d~ensions within the ranges of the variables con
sidered.
The network of points used in the numerical calculations was
.formed by two sets of parallel lines. The first set of lines drawn
parallel to the abutments divides the length of the span into eight
equal spaces. The second set of lines drawn parallel to the girders
divides each slab panel into two equal segments. To test the accuracy
of' the computed data using this network of points) analyses were made
for the right I-beam bridge (~ = 0), for which known exact solutions
are available for comparison. Comparisons of the exact and difference
solutions showed that the method of finite differences is ver,y satis
factor,y for the determination of' girder moments; even for a coarse net
work of points as used in this study, the agreement between the exact
and difference solutions is quite good. However, rather serious errors
are to be expected in the di.f.ference solutions for slab moments in the
floor system for certain load pOSitions, unless a much finer network
than used in this investigation is chosen. For the stru.ctures consid
ered, it was found necessary to apply certain corrections to some of
25
the influence values in order to make up for the coarseness of the net
work.
This investigation was made by T. Y. Chen and is reported in
his doctoral dissertation (Item 8 o.f Table I). This study was in part
supported by a different project. The results of the investigation
have been prepared for publication as a University of Illinois Engineer
ing Experiment station Bulletin which is now in press.
8. stress Analysis of Stiffened Shell structures
This investigation was concerned with the stress analysis of
stiffened shell type structures, such as are used in aircraft construc
tion. A simple numerical procedure was developed which requires but a
single viewpoint to handle the analysis of stiffened shells subjected to
both bending and torsion. This method is based upon the same simplifying
assumptions as are found in the previously developed analyses in this
field, but the procedure described herein is more general~ For example,
the procedure may be readily applied to the analysiS of shells with
unsymmetrical cross-sections and to multi-celled box structures.
In essence the analysis requires the solution of a set of
simultaneous linear equations in which the unknowns are the displace
ments at a discrete number of points in the structure. These equations
are never actually written out but instead are sol.ved indirectly by an
iterative procedure, working directly on a drawing of the structure.
Because the convergence of a straight iterative procedure is very slow
for this problem, a special method was developed for making a good
initial estimate of the displacements and also a special group correction
pattern was derived for occasional use to make large adjustments. The
iterative procedure is set up so that advantage is taken of the process
of continuous multiplication on an automatic calculating machine.
This investigation is reported in a Ph.D. dissertation by J. E.
Duberg (Item 12 of Table I).
9. ADalys is of Shells
Five investigations have been conducted with the object of
developing convenient numerical procedures for the analysis of various
shell problems. Two studies have been made on shells of revolution
loaded symmetrically, one study on the analysis of cylindrical roof
shells, one study on hipped plate structures, and one study of the
stresses caused by initial irregularities in tubes subjected to external
pressures. These investigations are discussed in the order named above.
(a) Pressure Vessel Heads
A numerical procedure for the analysis of pressure vessel heads
was developed by T. Au. Of course, the pressure vessel head is a shell
of revolution loaded with uniform internal pressure. The procedure dis-
cussed herein is based on the general theory of thin elastic shells due
to A. E. H. Love. The method is approximate in the sense that the govern-
ing mathematical equations are satisfied only in finite difference form.
However, it has' the merit of simplicity and of directly providing numeri-
cal values of normal forces, moments, and shears needed in design. The
appropriate finite difference equations were first developed and then
applied to the particular cases of pressure vessels baving spherical
27
heads~ conical heads, torispherical heads, toriconical heads, hemispher
ical heads, and flat circular heads. For the test problems considered
in this study, close agreement was obtained between the results found by
"classical" analysis and those obtained by the numerical method.
This investigation is reported in a Ph.D. thesis (Item 2 of
Table I) and in a technical report by Au, Goodman, and Newmark (Item 2
of Table n).
(b) Domes Under Symmetrical Loading
Another study of the analysis of' domes having the form of a
surface of revolution was made by R. Schmidt. One phase of this inves
tigation was concerned with studies of the application of the method of
finite differences and a second phase was concerned with the development
of' a procedure referred to as the method of finite elements. This method
is based upon considerations of the equilibrium and distortion of finite
size elements. The simplifying assumptions made in this procedure are
the same as those underlying A. E. H. Love's theory of shells.
The common feature of the methods used in this study is that
they reduce the problem of analysis of domes to the solution of a system
of s~ultaneous linear equations. Since, in order to achieve good accu
racy, a large number of equations is required, the use of' high-speed
electronic computers is indispensable for the majority of' problems. How
ever, with the rapid development of electronic computing, this is not
considered to be a great handicap.
In this study primary emphasis was placed on the elastic anal
ysis of domes of such proportions that the resulting deflections may be
28
considered to be SITIEtll in comparison vli th the thickness of the dome. How
ever, in addition) a procedure was suggested which makes it possible to
analyze domes exhibiting rnoderately large deflections.
This study is reported in a doctoral dissertation (Item 24 of
Table I).
(c) Cylindrical Shell Roofs
Cylindrical shell roof structures, often called barrel vaults,
are of great importance in civil engineering construction. A numerical
method for the analysis of such structures was developed by W. S. Schnobrich
and reported in his M.S. thesis (Item 23 of Table I).
In this method the shell is replaced by an analogous framework
\,fhich has a finite number of degrees of indetermw..acy. A distributed load
on the shell is transformed into equivalent concentrated forces acting on
the joints of the f!"a..rn.e1{Ork. The analogous frame\-Tork is analyzed by solv
ing a set of simultaneolls j linear algebraic equations involving the dis
placements of the joints as unknO\IDS. From the displacements the forces
in the framevork may be found readily. If the framework is suitably cho
sen, its behavior will closely approximate the behavior of the shell. The
displacements o.f the original shell will be approximately equal to the
displacements of the analogous framework at corresponding points, and
appro~{imate values of the direct stresses and moments in the original shell
rnay be found .from the forces and moments in the bars of the framework.
It is possible to approximate a cylindrical shell \-lith many dif
ferent types of framework arrangements. The framework arrangement devel
oped in this study consists of a system of rigid bars arranged in squares,
29
joined at their intersections by elastic joints, and internally linked
by torsion springs.
The value of the bar-spring system, like that of all numerical
procedures, lies in its complete versatility and adaptability. However,
the solution of a large number of simultaneous e'luations may be re'luired
to obtain good accuracy.
(d) Hipped Plates
In this investigation a moment distribution type analysis was
developed for structures composed of rectangular flat plate elements
connected side by side along their longitudinal edges and simply supported
on opposite edges. The method of analysis considers the flexural action
of the plate under normal components of loading, the plane stress action
o.f the plates under planar loads, and the effects due to joint displace
ments. The exterior longitudinal edges of the structure may be free,
simply supported, fixed, or elastically restrained.
The fundamental basis of this procedure is as follows. The
load is expanded into a sine series and each term is considered separately.
Due to each loading term it is assumed that all plates in the structure
deform in a sinusoidal shape in the longitudinal direction. In a manner
similar to Newmark's procedure for rectangular slabs continuous over
intermediate supports, the moments and direct stresses are found by a
moment distribution type, relaxation procedure. The procedure is not
exact but is based upon certain assumptions, especially with regard to
the distribution of the direct stresses in the structure.
30
This study is described in a Ph.D. thesis by G. S. Wu (Item 32
of' Table I).
(e) stresses Caused by Initial Irregularities
In a thin circular cylindrical shell subjected to uniform exter
nal pressure) only direct stresses are present in the center portion of
the shell. However, bending moments and corresponding stresses are intro
duced if the cross-section deviates from the circular shape due to
defects of manufacture) imperfect fabrication, improper handling, or the
action of external forces or shock. The purpose of this investigation
was twofold: (a) to develop a method of analysis of' the stresses caused
by an initial irregularity which is uniform along the length of the tube,
and (b) to determine the stresses in a cylindrical shell with a dent which
varies in magnitude along the longitudinal axis of the shell.
When the initial irregularity in a long tube is uniform along
its length} the moments, stresses, and deformations are also uniform and,
therefore) the stresses may be found from the solution of an analogous
two-dimensional ring. A numerical method of analysis was developed for
the determination of the stresses and deformations in rings with initial
imperfections and uniform external load. This procedure is a combination
of the stodola method of successive approximations and the numerical inte
gration method of Newmark. In this procedure both large deflections and
extensions of the center line of the ring are taken into consideration.
Due to the fact that the procedure involves a considerable amount of num
erical computation, a simplified method for predicting the maximum bend
ing moment developed in a ring subjected to uniform pressure was developed.
The simplified method is approximate) but it is direct. The results
obtained by these two methods check extremely well.
31
The second objective achieved in this investigation was the
determination of the stresses in a thin cylindrical shell with a dent
varying in magnitude along the longitudinal axis of the cylinder which
is subjected to uniform pressure load. The dent is assumed to have
the shape of one of the buckling modes on a cross-section and to vary
longitudinally as a sine wave. By assuming that the initial irregu
larity is small and that its square can be neglected, all fundamental
quantities such as curvatures, strain and stress components, which gen
erally have very complicated mathematical expressions, are expressed
in comparatively simple forms. Three governing displacement equations
of equilibrium for the case of a circular cylindrical shell under uni
form pressure are obtained. They contain correction terms due to the
assumed initial irregularities.
This investigation is described in a doctoral dissertation by
T. S. WU (Item 33 of Table I) and in a technical report by Wu, Goodman,
and Newmark (Item 14 of Table II).
32
CHAPI'ER III
INVESTIGATIONS IN BUCKLING AND VIBRATIONS
In this chapter are described a variety of investigations in
buckling of plates and beams, and in steady-state forced vibration and
free vibration of elastic systems of various kinds.
10. Numerical Procedures for Vibration Problems
Two of the most powerful numerical methods for solving problems
of free vibration and steady-state forced vibration are: (a) the Stodola
Vianello method of successive approximations, and (b) the step-by-step
method developed by Holzer.
A review of these methods with some generalizations and appli
cations to a variety of problems was made by S. Aisawa who developed a
convenient scheme for arranging the computations for both methods. This
scheme is similar to that developed by Newmark and originally used in
connection with the computation of moments, deflections, and buckling
loads of bars.
Both methods were found to be well suited for the determina
tion of the natural frequencies of spring-mass systems having several
degrees of freedom. Studies were made of three different procedures for
evaluating the higher natural frequencies and modes by the method of
successive approximations. For the analysis o.f the steady-state forced
vibrations of simply-connected spring-mass systems, the step-by-step
method was found to be more advantageous than the successive approximations
procedure. The step-by-step method was extended also to the case of vis
cously damped systems.
33
The results of this study have been reported by Aisawa in his
M. S. thesis (Item 1 of Table I).
A second investigation concerned with the solution of charac
teristic value problems in general, and specifically with the determina
tion of the natural frequencies and modes of vibration of beams, \vas
conducted by Mlfron L. Gossard. Dr. Gossard demonstrated the use of the
so-called iterative transformation method.
In NACA Report Noo 1073, Gossard describes as follows the gen
eral £eatures of this method: ~e principle of the iterative transfor
mation procedures is similar in form to that of the standard iteration
procedure for solving characteristic-value problems. Both procedures
require the determination of the solutions in the order of the magni
tudes of the eigenvalues, beginning with the f'undamental. Both proce
dures require assumptions of modes, integrations which generally must
be done numerically, and sweeping operations for higher-order-mode deter
minations. The distinguishing features of the iterative transformation
procedure occur in the determination of solutions higher than the funda
mental and are as follows g (I) The immedia te aim is to determine not the true
nth mode, as in the standard iteration procedure, but a particular linear
combination composed of all modes from the fundamental to the nth 0 This
linear combination is referred to as the transformed nth mode. The
transformed nth mode can be made to have nodal (zero) points at specified
stations of the wing; such a feature is highly desirable in numerical
work 0 (2) The sweeping operations, which consist of subtractions of
lower-order-mode shapes from the function obtained by integrating the
34
assumed mode, do not employ the orghogonality relations as in the stand-
ard iteration method but make use of forcing functions that, in numerical
work, greatly simplify the sweeping operations and increase the over-all
accuracy of the results by making the sweeping operations more consistent
with the rest of the process. (3) Although the true nth eigenvalue is
determined directly in the iterative transformation procedure, the true
nth mode must be computed from quantities within the iteration cycle
after the transformed nth mode is found."
The application of the iterative tr~nsformation procedure to
the problem of calculating modes and frequencies of natural vibration of
beams in flexure was first described by Gossard in his doctoral disserta-
tion (Item 14 of Table I).
ll. Natural Frequencies of Elastically Restrained Bars
The transcendental equation for the natural frequencies of elas-
tically restrained bars is well known. However, this equation is not
only difficult to evaluate but also it does not exhibit the relative
importance of the various parameters involved. The purpose of this inves-
tigation was to develop a s;mple approximate procedure for determining
the natural frequencies of elastically end-restrained prismatic bars.
The restraints may be due to elements such as coil springs or they may
result from the continuity of the bar with adjoining members.
The natural frequencies of the elastically restrained bar have
been expressed as the product of an end-fixity coefficient C multiplied n
by the fundamental frequency of the same bar simply supported at the
35
ends 0 From the results of numerical calculations based on the exact
solutions a simple empirical formula was developed for the coefficient
c. This formula is applicable only to bars for which the stiffnesses n
of the end restraints are positive. The formula is valid for the fun-
damental as well as the higher natural frequencies and is accurate to
within a maximum error of' 4 per cent, the max:ilIlum error occurring in
the fundamental frequency.
The results of this study can be used to calculate also the
fundamental frequencies of two-span beams and of particular arrangements
of three-span beams continuous over non-deflecting supports and elasti-
cally restrained against rotation at the ends. As before, the stiffnesces
of the end restraints are assumed to be positiveo
The problem of determining the natural frequencies of a con-
tinuous beam is basically the same as that of determining the frequencies
of one of its spans only, provided proper account is taken of the actual
restraints existing at the ends of the Single spano A procedure was
developed which consists of progressively reducing a continuous beam to
its elastically restrained, one span analogue. In this procedure the
dynamic end restraints are expressed as the product of the restraints
which are provided under static conditions multiplied by a diminution
factor which takes account of the inertia effects. Once this one-span
analogue is defined, its fundamental frequency, and therefore the fun-
damental frequency of the original continuous beam, is evaluated from
the approximation formula referred to previously.
I I 1 I
This investigation bas been described in two technical reports
by Newmark and Veletsos (Items 5 and 18 of Table II) and in two pub
lished papers by the same authors (Items 10 and 13 of Table III).
12. Vibration of Continuous Flexural Systems
This investigation was concerned with the free vibration and
the steady-state forced vibration of continuous flexural systems having
distributed mass and elasticity. A method was developed for calculating
the undamped natural frequencies and the corresponding natural modes of
cont~uous beams on rigid or flexible supports, continuous frames with
out sidesway, symmetrical single-balf multi-story frames for which the
joints are free to translate, and continuous plates having two opposite
edges simply supported. The method can be used also to determine the
steady-state forced vibration of these systems. A system with distri
buted mass and elasticity has an infinite number of natural frequencies.
With the method developed one is capable of determining as many of the
natural frequencies as he desires.
The method is a generalization of the well known Holzer method
for calculating natural frequencies of torsional vibration of shafts.
Like Holzer's method, it has been reduced to a routine scheme of compu
tations which, when repeated a sufficient number of t~es will give the
natural frequencies of the system to any desired degree of accuracy.
The method is based on the fact that, in the absence of damp
ing, the exciting force (the term force is used in a generalized sense
to indicate a force or a couple) which is necessary to maintain a
37
dynamical system in steady-state forced vibration with finite amplitudes
becomes equal to zero at a natural frequency. Briefly, the method con
sists of (a) assuming a frequency of vibration, (b) determining the mag
nitude of the exciting force which when applied at some appropriately
selected joint of the structure will produce a vibration configuration
having a fixed amplitude of displacement at some other joint, (c) repeat
ing these steps for a number of frequencies, and (d) plotting the excit
ing force as a function of the frequency of vibration. The frequencies
for which the exciting force vanishes represent the desired natural fre
quencies of the system.
For an assumed frequency of vibration, the magnitude of the
exciting force can be determined by a number of different procedures.
The conditions to be satisfied are simply those of equilibrium and con
tinuity for each joint of the system considered. To satisfy the condi
tion of equilibrium, the sum o.f the moments and forces at the ends of
the members meeting at a joint must be, respectively, equal to zero. To
satisfy the condition of continuity, the slopes of the members meeting
at a joint must be equal and also the deflectionsof' the members meeting
at the joint must have the same magnitude. These conditions may be
expressed in equation form in a number of different ways and the equations
may be solved by a number of procedures. In the method adopted, these
conditions are expressed in the form of a generalized slope-deflection
equation, and the distortions of the structure and the exciting force are
computed by the repeated application of this equation, working progres
sively from one end of the structure to the other.
Extensive tables of numerical values have been computed for
the various quantities which are necessary in the analysis by this method.
With these tables the calculations required in the application of the
method. to particular problems are sim:plified greatly. The tabulated
values may be used also with other analytical techniques as well as for
the analysis of the steady-state forced vibration of structureso
This study was made by A. S. Veletsos and is reported in his
doctoral dissertation (Item 31 of Table I). The results have been
summarized in three technical reports (Items 9, 16, and 20 of Table n)"
in three published papers (Items 12, 15, and 11 of Table ill), and in
a discussion of a paper (Item 21 of Table III).
13. Effects of Rotatory Inertia and Shearing Distortion on Vibration of' Bars
The natural frequencies of beams as predicted by the classical
Bernoulli-Euler theory are known to be higher than those obtained in
care~ controlled experiments. The difference between theory and
exper~ent is due mainly to the fact that the classical theory does not
consider the influence of the rotatory inertia and shearing distortion.
This difference may become practically significant when the cross-sec-
tional dimensions of the beam are small compared to the length of the
beam between nodal sections.
The first correction to the classical equation of motion was
made by Lord Rayleigh. Recognizing that the elements of a vibrating bar
perform not only translatory motion but also a rotatory motion, Rayleigh
included the inertia load due to the rotatory motion and studied its
39
influence on the response of the beam. Later, Timoshenko showed that a
still more accurate differential equation of motion can be obtained if
one includes also the influence of the deflection due to shear.
In the present study Timoshenko's differential equation was
derived starting from the equations of motion of the theory of elastic-
ity. Proceeding in this way, it is possible to exhibit the importance
of the various approximations and parameters used for purposes of sim-
plification. A procedure was developed for estimating the magnitude of
the shear deflection coef.ficient, and the values of this coefficient
were computed for various cross sections. These results, partly original
and partly due to other investigators, were then used to calculate the
natural frequencies of simply supported and cantilever beams.
This study was conducted by J. G. Sutherland and L. E. Goodman
and was described in a technical report
discussion of a paper (Item 2D of Table III).
140 Vibration of an Elastic Solid
This investigation was concerned with the free vibration of an
elastic solid bounded by two parallel planes. It was found that two
fundamental radially symmetric solutions of the linear theory of elastic-
ity describe the free vibrations of an elastic plate of finite thickness.
These solutions correspond to "extensional" and "flexural" motions and
are analogous to the plane strain waves discovered by Lamb and Rayleigh.
Following the treatment of Lamb, the transcendental equation
relating the frequencies of vibration to the wave shape were developed,
40
and the values of the frequencies were evaluated as a function of the
wave length for the fundamental and for the higher modes. From a study
of the higher modes of vibration a family of surfaces was found across
which no stress is transmitted. These surfaces define a class of solids
of revolutionfoT which the motions in question represent normal modes
of' free vibration.
For very thick plates the solutions developed correspond to
Rayleigh surface waves" while" for thin plates, they approach asymptoti-
cally, well known results of the elementary plate theory.
This investigation was conducted by L. E. Goodman and was des-
cribed by him in a paper (Item 8 of Table III) which was reprinted and
is sued as a technical report (Item 13 of Table II).
15. Buckling of Elastically Restrained Non-Prismatic Bars
This investigation was concerned with the elastic buckling of
bars for which the moment of inertia of the cross section varies linearly
from a minimum at the ends to a maximum at midspan. It is assumed that
the bars are compressed by concentric end forces" that they rest on non-
deflecting supports and are elastically restrained against rotation at
the ends.
The characteristic equation for the critical buckling load was
derived and was used to evaluate the lowest critical loads for various
degrees of end fixity and various ratios of I /1.. The buckling max nun
load for the elastically restrained bar was expressed as the product of
an end fixity coefficient C multiplied by the corresponding buckling load
41
for a bar hinged at both ends. From the results of numerical calcula
tions based on the exact solution empirical equations were next developed
for the coefficient C and the buckling load of a bar hinged at both ends.
The maximum error in these equations is of the order of 2 1/2 per cent.
The equation for C is valid only for positive end restraints.
The strength of bars with linearly varying "I" was finally
compaxed with that of prismatic bars having the same volume. In general,
hinged bars or bars subjected to moderate end restraints are stronger
than ~rismatic bars of the same volume. However, fixed bars and bars
subjected to large end restraints are weaker than prismatic bars of the
same volume.
The results of this study were summarized by A. S. Veletsos
in his M. S.. thesis (Item 30 of Table I)"
16. Buckling of Plates
A study of numerical procedures for problems of plate buckling
was made under the sponsorship of this program. A relaxation-iteration
type procedure was developed for solving the homogeneous linear, simul
taneous equations which result from the use of finite difference approxi
mations to the pertinent differential equations. A study was made of
methods to speed up the convergence of the calculations. This feature
has considerable importance if electronic computing machinery is not
available.
A special procedure was developed for cases where Levy's solu
tion may be used to reduce the partial differential equation to an
42
ordinary differential equation. For this case a solution may be obtained
very easily. Of course, most analytical solutions fall in this class.
The problem of buckling in the large deflection range bas been
treated very briefly. A numerical procedure was developed for solving
the two simultaneous, non-linear partial differential equations by means
of .finite differences. However, the solution is quite laborious for
hand computation.
The stUdies described above are reported by W. J. Austin in a·
doctoral dissertation (Item 3 of Table I).
After the thesis was completed an improved method was developed
for the solution of plate buckling problems under conditions for which
the usual differential equation can be reduced to an ordinary differen
tial equation. This method is a combination of the Stodola method of
successive approximations with the numerical procedure of integration by
Newmark. Plates reinforced with rib stiffeners were considered as well
as unreinf'orced plates. The general procedure developed in this work is
applicable to many other problems, since it permits a relatively s~le
and accurate numerical integration of a class of differential equations.
This study is published in a technical report by Austin and NeWIIJark (Item
1 of Table II) and in a paper (Item 5 of Table III). The paper was
reprinted and issued as one of three papers in a technical report by
Newzna.rk, D' Appolonia, and Austin (Item 12 of Table II).
17. Torsional-Flexural Buckling of Beams and Columns
Four studies have been made on the problem of torsional-flex
ural buckling of beams and columns. The first two stUdies were concerned
43
primarily with the development of simple numerical procedures and the
last two studies were concerned with solutions of practical problems of
importance.
(a) Development of Numerical Procedures
The first study was concerned with the development of a numeri
cal procedure for determining the critical torsional-flexural load of
beams and columns 0f open section and with an investigation of the effect
of web deformations on the torsional buckling strength of I-section col
umns with thin webs.
The procedure developed is of the Stodola type of successive
approximations. In this procedure the necessary integrations are per
formed numerically using first Newmark's integration technique and then
an initial value type, step-by-step procedure. The solution converges
quickly to the lowest critical load.
The study of the effect of deformation of the cross-section
during buckling on the torsional buckling load of columns of T-section
was carried out by means of the calculus of variations. Two distorted
configurations of the web of the twisted I-section column were assumed
and the minimum load required to maintain each in its buckled configura
tion was found. One consideration leads to the critical stress produc
ing torsional buckling and web deformation. The other yields the cri
tical stress which causes local failure in the web accompanied by
twisting of the flanges without rotation of the cross-section as a whole.
It is called ~uckling by ,flange twist. It The critical loads of several
columns were computed and compared with the critical loads given by
44
Euleris and Wagner's theories. In general, it was found that the reduc
tion in the torsional buckling strength of columns due to deformations
of the web is negligible in the case of I-sections with proportions
commonly used in engineering practice.
This investigation was conducted by MOunir Badir and is reported
in his Ph.D. thesis (Item 4 of Table I).
A second numerical procedure for the solution of torsional
flexural problems was developed by C. P. C. Tung and is reported in a
doctoral dissertation (Item 28 of Table I). This procedure is a combina
tion of the Stodola method of successive approximations and the numeri
cal procedure by Newmark. This procedure is more versatile and more
accurate than that developed by Badir, but is subject to convergence dif
ficulties in some cases. The procedure is illustrated for problems of
lateral buckling of beams, torsional buckling of colUIlllls, and lateral
torsional buckling of beam-columns due to combined transverse and longi
tudinal loads. In addition to the development of the analytical proce
dure, ~. Tung worked a series of important practical problems and in
his dissertation he discusses these solutions as they relate to current
steel design formulas.
(b) Elastically End-Restrained I-Beams
The design of beams subject to lateral buckling is based upon
formulas for the critical stress of simply supported beams which are
uniformly loaded on the top flange. In a large investigation at the
University of Washington these formulas were verified by experiments
conducted on beams with simple supports. In addition, some tests were
also run on beams with more practical end conditions, such as clip angles
or seat connections" and it was found that the restraint offered by these
simple connections increased considerably the ultimate load.
The purpose of this investigation was to provide a theoretical
stu~ of the critical loads and corresponding stresses for I-beams with
rotational, elastic end restraints. In practical structures these
restraints are, of course, provided by the supports. In this study the
beam was assumed at both ends to be elastically restrained against rota
tion about both principal axes and to be held against rotation about
the longitudinal axis. Both uniform load and concentrated load were
considered, with the load assumed to be applied at the top flange, the
centroid, and the bottom flange. Fifteen hundred solutions were made
for the practical range o.f the variables. The solutions were made numer
ically using finite differences. The solutions were obtained on the
ILLIAC in about 10 hours 0
The solutions have been presented in the form of tables and
charts in a paper by Austin, Yegian, and Tung (Item 14 of Table III).
( c) Beams Supported by Cables
The purpose of this investigation was to find the critical
weight O.r slender girders when suspended in the air during erectio?- pro
cedures. The girders are assumed to be lifted by two vertical cables
attached by clamps to the top flange. The two lifting cables may be
attached at the ends or they may be at some point inside so that the
beam is divided into a central span with overhangs at each end.
The critical weight of I-beams, both bi-symmetrical and mono
symmetrical, were found for a wide range of parameters covering all
practical conditions. In addition, a study was made of the maximum
stresses which occur in bi-symmetrical I-beams when the cables are
attached with small initial eccentricities. The study of stresses was
made to determine the effect of small deviations from idealized con
ditions on the ultimate strength. The solutions were made numerically
on the ILLIAC. Two analytical procedures were used, a successive
approximations and integration procedure, and a finite difference
method.
The solutions, in the form of tables and curves, and inter
pretations of the solutions in practical terms are presented in a
thesis by S. Yegian (Item 34 of Table I). It is planned to prepare
this work for a technical report and formal publication.
This investigation \iaS begun under the sponsorship of Task VI
and has been concluded under sponsorship of Contract Nonr-1834(03), Task
III.
47
CHAPrER IV
RESPONSE OF SERUCTURES UNDER TRANSIENT LOADING
The investigations described in this Chapter concern the
development of numerical procedures for structural dynamics problems
and the subsequent application of these procedures and of other infor
mation in the solution of important practical problems connected with
the response of structures to earthquake and blast loading.
18. Development of Numerical Procedures of Analysis
There have been developed under this program several numeri
cal procedures .f'or the analysis of the response of' structures sub
jected to dynamic loading. An objective held constantly in view during
the course of the investigations has been to develop procedures which,
with a reasonable amount of' work, can give approximate answers suffi
ciently accurate for engineering purposes and which, with additional
work, yield results to any desired degree of accuracy. Attention has
been focused on simplicity of concept and generality of' procedure.
The procedures developed in this investigationJ which are des
cribed in more detail in the following articles, have several features
in common. First, the procedures were derived pr:il:narily for structures
which are considered to consist of separate concentrated masses sup
ported on a flexible but weightless system of framing. As in an actual
structure the mass and elastiCity are usually not concentrated but dis
tributed, the structure which is analyzed is only an approximation to
the actual structure. But with sound judgment a reasonable approxima
tion can generally be rrBde so as to result in fairly good accuracy in
the calculations.
Second, the procedures described here all involve a process
of step-by-step integration. That is, the time coordinate is first
divided into a number of short intervals. The acceleration, velocity,
and displacement of a mass at the beginning of any particular time
interval are either given or known from previous histo~J of the struc
tural behavior. From these are computed the acceleration, velocity
and displacement of the mass at the end of the interval by devices
which differ for different procedures and which distinguish one pro
cedure from another. I)y repeating this process of integration succes
sively for all the intervals of time within the range of interest, a
complete solution for the dyrilimic response of the structure is obtained.
Any type of loading such as that due to blast of a bomb,
impact from a moving object, or foundation motion due to an earthquake,
can be considered.
(a) Procedure Using TaylorYs Expansion
In this procedure the displacement and velocity of the mass at
the end of the time interval are computed by means of a Taylor~s series
whose terms involve the displacement and its time derivatives at the
beginning of the interval.
The procedure was developed primarily for the study of flex
ural vibration of beaw~, but it applies to any transient problem for
which the governing e~uation of motion is linear and has constant
coefficients. \thile only simple problems of vibration within the elastic
region have been considered, the procedure may be extended to problems
involving plastic action, to beams resting on elastic or rigid supports,
to simply-supported or continuous spans, to beams subjected to axial as
well as lateral loads, and to cases with damping.
Several examples were solved by this procedure and conclusions
were deduced from comparisons with exact solutions and solutions obtained
by other methods.
The details of this procedure were presented in full in a Ph.D.
thesis by Z. K. Lee (Item 19 of Table I).
(b) Newmark's a Method
This is a generalized method of numerical integration which
emphasizes the physical aspects of the problem. For each time interval
considered the solution is carried out by an iterative scheme. A trial
value is estimated for the acceleration of each mass at the end of the
time interval. From this value and from the values of the accelera-
tion, velocity, and displacement at the beginning of the time interval,
the velocity and displacement of each mass at the end of the time inter-
val are computed by means of appropriate quadrature formulas. From
these quantities new accelerations at the end of the interval are evalu-
ated by use of Newton's second law of motion applied to each mass. If
the derived accelerations do not agree with the assumed values, another
trial is made using the derived values as the initial assumed values.
The procedure is repeated until derived and assumed values agree within
a desired accuracy.
50
The quadrature formulas used involve a quantity ~ which defines
the nature of the variation of the acceleration within the time interval.
By a proper choice of ~, the method reduces to any of a number of clas
sical numerical procedures. For example, the method reduces to the con
stant acceleration method for ~ = 0; to Newmark's linear acceleration
method for ~ = 1/6; to Timoshenko's modified method for ~ = 1/4; and to
FOx's method for ~ = 1/12.
Two important aspects of this work were concerned with (a)
the convergence of the successive iterations for each time interval and
(b) the evaluation of the errors inherent in the numerical procedure.
Particular attention has been given to defining the conditions under
which the various procedures will either fail to converge or lead to
exceedingly inaccurate (unstable) solutions.
The method may be used to determine the dynamic response of
structures having any relationship between force and displacement ranging
from linear behavior through various degrees of inelastic action up to
failure. The same general techniqUe may be uSed also when damping is
present.
A complete discussion of this method has been given in a paper
by N. M. Newmark (Item 4 of Table III). This paper has been issued also
as a technical report (Item 10 of Table II).
(c) ~tension of ~ Method - Use of Dynamic Load Factor
This procedure is an extension of the ~ method and represents
an attempt to improve the accuracy of the ~ method for certain condi-
tions. The method of analysis is exactly the same as the p method,
51
except that here the external loads are modified by selected factors in
the course of the computation.
The derivation of the expressions for the load factors and
the illustration of their application to several examples have been
included in an M.S. thesis by T. C. Hu (Item 16 of Table I). It was
found that this method of dynamic load factors has merit under the
following conditions:
(1) An approximate value of the fundamental natural fre
quency of vibration of the system is known.
(2) In a given time interval the loading either is discon
tinuous or lasts over a portion of the time interval.
(3) The maximum response of the structure occurs at the
primary stage of the loading.
19. Review of Numerical Procedures of Analysis
There are as many different procedures for the numerical anal
ysis of dynamic structural response as there are methods for the numeri
cal integration of differential equations. The objectives of this inves
tigation were to stud~ the accuracy and ran~e of applicability of various
methods of numerical integration now available and often used in dynamics
problems, and to obtain data that will enable one to make a judicious
choice of a suitable technique for a specific problem at hand.
Two systematic and fairly comprehensive studies have been made
and the conclusions have been summarized in two technical reports, the
first by Newmark and Chan (Item 7 of Table n) and the second by Tung
52
and Newmark (Item 19 of Table II). The first study is also reported in
the doctoral dissertation of S. p. Chan (Item 7 of Table I). In both
studies comparisons have been made between the various techniques with
respect to the accuracy of solutions at any stage of the integration
process, the nature of errors, the limitations imposed by considera
tions of stability and convergence, time consumption, and self-checking
provisions of the procedures. The studies have been confined to a
single-degree-of-freedom system. However, the comparisons made can be
used equally well for more complicated systems, since the motion of
multi-degree-of-freedom systems can be considered as being made up of
the motions of the eigenmodes, each mode acting as a single-degree
of-freedom system.
The methods considered include the constant acceleration
method, Timoshenko:s modified acceleration method, Newmarkis ~ methods,
the various methods of finite differences due to Levy, Salvadori,
Houbolt, Adams and others, Euler's and modified Euler's methods, the
third order rule of Runge-Heun-Kutta, Kutta's fourth order rule, and
methods involving Taylor expansions.
One of the conclusions drawn from these stUdies is that the
~ method is particularly well suited for the analysis of dynamic struc
tural response because or its simplicity and rlexibility in applica
tion. Other significant conclusions may be found in the reports cited
above.
53
20. Response of structures to Earthquake Motions
Present methods of earthquake resistant design of structures
are mainly based on engineering judgment, experience, and practice.
The studies reported herein are intended to furnish information which
will help to remove some of the indeterminaGY associated with the
determination of the magnitude of the forces for which provision must
be made in design. These studies are sUlll1IlB.rized in the following
articles. It should be noted that these studies have a wider field of
application than the field of earthquake resistant design. With proper
modification they can be used to predict the response of other dynami-
cal systems subjected to "ground" disturbances which possess cbaracter-
istics of randomness.
(a) Influence of Ductility on the Response of Simple structures to Earthquake M::>tions
The object of this investigation was to estimate the influence
of ductility on the ability of steel frame mill buildings to withstand
earthquakes. A simplified one-story steel frame symmetrical about its
center line was considered. It was assumed that the mass of the struc-
ture was concentrated at the tops of the columns and that the roof
truss was infinitely rigid as compared to the colunms. The structure
may then be analyzed as a system with a single-degree-of-freedom. The
material of the structure was assumed to have an idealized elasto-plastic
behavior, that is, a "flat-top" stress-strain, or load-deformation curves
The studies made by G. W. Housner show that the records of
past strong-motion earthquakes exhibit characteristics of randomness
54
and that in so far as their effect on the response of structures is con
cerned, the actual earth~uakes may be replaced by a set of fictitious
random ground motions. Based on this principle, five possible ground
motions were constructed for the purpose of this investigation. Each
of the motions considered consisted of 20 stepped acceleration pulses
which were selected so as to be random in direction, in amplitude, and
in duration.
The principal parameters considered were~ the natural fre-
quency of vibration of the st~~ctvIe; the root mean square amplitude of
the ground motionj the maximum dynamic deflection of the top of a col
umn with respect to the basej and the maximum deflection of the top of
a column relative to its base which can take place before the onset of
plastic action in the column. The last quantity is a measure of the
ductility of the structurej its value is infinite for a perfectly elas
tic structure and zero for a perfectly plastic structure.
A graphical method of analysis, similar to the gyrogram or
phase-plane diagram, was developed for the response of an elasto-plastic
single-degree-of-freedom system subjected to a complex base motion.
The method may be extended to include the effect of strain-hardening by
the introduction of an additional parameter. The application of the
method is not restricted to structures subjected to stepped acceleration
pulse functions as considered in this study; any ground motion for which
the acceleration is a linear function of time may be handled with equal
ease.
55
Using this method, the responses of a number of structures
covering a wide range of the parameters described above were deter
mined for each of the five random ground motions considered. From
these data conclusions were drawn regarding the effect of ductility
on the maximum response of structures due to earthCluakes.
This study is described in a Ph.D. thesis byS. L. Pan (Item
20 of Table I) and in a technical report by Pan, Goodman, and Newmark
(Item 3 of Table n).
(b) Aseismic Design of Elastic structures Founded on Firm Ground
The object of this investigation was to arrive at a rational
basis for the design of earthquake resistant structures. The scope of
this work was limited to structures whose behavior is essentially elas
tic and which are founded on firm ground. The structures may have any
number of degrees of freedom, and both structure and ground may have
viscous damping.
The method which has been developed is based on inferences
drawn from accelerograms of strong-motion earthquakes and on the theory
of probability. structures designed to withstand the responses com
puted by this method will be of uniform strength in the sense that all
their parts will have equal probability of successfully withstanding
an earthquake. Different parts of a structure will not be relatively
overdesigned or underdesigned.
While the sources of excitation considered in this study have
been ground tremors, they may, with slight changes in the analysis, be
wind gusts or other dynamic disturbances.
This investigation is described in a doctoral dissertation by
E. Rosenblueth (Item 22 of Table I), in a technical report by Goodman"
Rosenblueth, and Newmark (Item 6 of Table II), and in a published paper
by the same authors (Item 11 of Table III).
(c) Response of a Typical Tall Building to Actual Earthquakes
The ~ic response of a 10-story structure subjected to the
action of strong-motion earthquakes has been evaluated with the aid of
the ILLIAC. The ground motions considered included twelve different
strong-motion earthquakes recardedby the U.S. Coast and Geodetic Sur
vey. The structure was assumed to react as an elastic shear-beam. Two
dif.ferent configurations of the building were considere~ (a) a building
with uniform distribution of mass and stiffness, and (b) a building with
a linear distribution of mass and stiffness.
The calculations were performed for three different damping
conditions corresponding to (a) no damping; (b) 2 per cent of critical
damping; and (c) 10 per cent of critical damping. The damping coefficient
was computed on the basis of the fundamental frequency and average weight
of the building. In these analyses each accelerogram was approximated
by a series of polygonal lines. The time interval used in the numerical
integration process was 0.0075 sec. Each problem required from 20 to 30
minutes of machine time with about one third of the time used in punching
results on the output tape.
The magnitude and distribution of the maximum dynamic shears
throughout the building were evaluated and the results were compared with
those obtained on the basis of the procedure recommended by the Joint
57
Committee on Lateral Forces. The maximum shears determined by these elas-
tic analyses are found to be significantly larger than those predicted on
the basis of the design recommendations of the Joint Committee on Lateral
Forces.
This study has been described by Tung and Newmark in a techni-
cal report (Item 25 of Table II) and in two papers (Items 18 and 19 of
Table III).
( d) Distribution of Extreme Shear in a Tall Building Subj ect to Earthquakes
This study was made to determine statistically the relative
distribution of the extreme shear developed at different stories of a
tall building which is subjected to the shocks of an earthquake. Such
knowledge is helpful in achieving a unified aseismic design of tall
buildings.
For the purpose of analysis) a tall uniform shear beam was
assumed as an idealized model of a tall building frame. An obvious
advantage of this simulation lies in the simplification obtained in
the governing equation of motion which is the one-dimensional wave equa-
tion. The solution to this equation may be expressed in two parts) one
representing a forward wave starting at the base and propagating toward
the top) and the other representing a wave traveling from the top to
the base) or a reflected wave. Both parts are functions of the velocity
of propagation of the disturbance along the beam.
For making a statistical estimate) the ground motion was
assumed to consist of a large number of random pulses) each having an
t,
equal order of magnitude. The state of shear in the structure due to
the random pulses is that produced by the shear waves traveling from
base to top and reflecting back to the base. An analogy may then be
drawn between this problem and the problem of random walks in which a
particle may move forward or backward, or may stand still on a line,
according to specified probabilities.
Based on this principle and the use of statistical methods,
a frequency distribution function for the story shear developed at
different levels of the structure was determined. The results obtained
indicate that the dynamic story shear which is a random variable in the
present analysis, possesses a normal distribution with a zero mean, and
that the ratio of the expected extreme shear at any story to the maxi
mum base shear varies parabolically with the height of the story from
the base.
This study is described in a technical report by Tung and
Newmark (Item 26 of Table II)o
21. Response of structures to Blast Loading
The field of design of structures to withstand the shock waves
produced by the detonation of high explosive and atomic bombs is rela
tively new and is one in which available unclassified literature has
been sparse. While an increaSing number of analytical techniques have
been developed for the determination of the responses of structures to
blast loading, there still remain many practical problems of design on
which information is lacking. These problems are of fundamental impor
tance from the designer's point of view.
The aim of this phase of the project therefore has been to
obtain pertinent data which may serve as a basis toward establishing
rational design criteria for structures subjected to blast loading.
59
Work accomplished along this line includes~ (1) a study of the response
of simple systems to dynamic loads and of the effect on that response
of variations in the characteristics of the excitation and in the prop
erties of the structure; (2) an investigation of the blast loading
transmitted from wall coverings to the building frames; and (3) studies
of the effect on dynamic response of foundation coupling and vertical
loads- In addition, an engineering design procedure for blast resist
ant structures was developed. These studies are described briefly in
the following articles. In general, only the simplest typesof struc
tures were considered, that is, those which may be simulated by single
degreee-of-freedom systems. While the conclusions made in these studies
pertain only to simple structures, it is believed that the general trend
of the results obtained will throw light on the dynamic behavior of more
complicated structures as well.
A general and concise discussion of the .fundamental principles
and various aspects of dynamic analysis and design has been presented in
the paper "Analysis and Design of structures Subjected to Dynamic Load
ing" by N. M. Newmark (Item 3 of Table In).
( a) Response of Simple structures to Dynamic Loads
The objectives of this investigation were (a) to study the
response of simple structures to the action of dynamic forces such as
those arising from the explosion of bombs, and (b) to determine the effect
60
on that response of variations in the valuesof each of the parameters
entering into the problem. The importance of the second objective
becomes apparent when one recalls that properties of a structure and
the characteristics of a dynamic disturbance are seldom known accu
rately. In order to interpret the significance of calculated responses,
it is essential for a designer to know to what extent variations in the
fundamental physical parameters may affect the response of the struc
ture.
This study was restricted to the action of undamped systems
possessing a single degree of freedom. It was assumed that the struc
ture can deform elastically up to a point and that for deformations in
excess of this limit it behaves in an ideally plastic manner. The types
of forces considered included the rectangular pulse and triangular
pulses with their peak value at the beginning, the end, or the middle
of the interval of time over which the disturbance is spread. Exten
sive numerical calculations have been carried out to determine the maxi
mum response of the structure, and the results obtained have been sum
marized in a set of useful design charts. Also, graphs have been pre
pared for the effect upon the maximum response of small variations in
the magnitude of each of the quantities entering into the problem. With
these charts, the determination of the maximum response of a structure
subjected to a blast load and the sensitivity of the response to changes
in the magnitude or shape of the pulse, or in the characteristics of
the structure, may be determined readily.
61
From the results of these studies, a simple semi-empirical
formula was developed by Newmark for the maximum response of a single
degree-of-freedom elastoplastic system subjected to a triangular pulse
with an initial peak. The accuracy of this formula has been tested
for a wide range of the variables involved and it has been found to
be extremely satisfactory.
This study is described in a M.S. thesis by N. B. Brooks
(Item 6 of Table 1) and in a technical report by Brooks and Newmark
(Item 15 of Table II).
(b) An Engineering Approach to Blast Resistant Design
Nearly all previous studies on the subject of the resistance
of structures to atomic blast have been concerned primarily with the
analysis of the response of a particular structure to a given loading.
However, this approach is at best of academic interest only and is
not very satis.factory from a designer's point of view, because the
structure to be analyzed does not yet exist, and the real problem that
confronts the designer is the preliminary choice of the structure.
Furthermore, many uncertainties exist in the details of the blast load
ing and in the properties of the structure. It can be demonstrated
that even minor variations in these details may cause major changes in
the computed responses. Elaborate analytical techniques are time con-
snming and mRy even be misleading if the results are applied indiscrimi-
nately to practical designs.
On the other hand, a procedure which involves rather crude
approximations but permits the designer to arrive at a preliminary
choice of sections directly, intelligently and fairly rapidly, should
prove of great ~practical value. The purpose of this study was to develop
such a procedureo
The procedure developed is believed to be particularly useful
for preliminary design of a new structure or for strengthening an exist
ing structural frame or its component parts a In this procedure one
needs to est:ima.te (a) the natural period of vibration of the frame or
component parts in question relative to the duration of the loading on
the structure, and (b) the ductility factor, defined as the ratio of
the permissible limiting de.flection to the yield-point deflectione
Using these est:ima.tes and certain approximate relations which have been
developed, one can deter.mine readily the required yield resistance of
the structure for a specified pressure. In essence this procedure leads
to the computation of either the yield-point resistance of the structure
necessary to resist a given overpressure or, alterna~ively, the peak
overpressure which can be resisted by a given structure.
This study is described in a paper by Newmark (Item l6 of
Table III). This paper has been reprinted and issued as a project tech
nical report (Item l7 of Table I1)0
(c) Blast Loading Transmitted from Walls to Building F1rames
When a structure having wall coverings is subjected to blast
pressure from the detonation of bombs J the load that is transmitted
from the walls to the structural frame may have a pulse-time relation
quite different from that of the pulse which impinges on the walls. It
was the purpose of this investigation to study the load-time relation-
ship of the reaction exerted on the frames 0
A typical bay of a one-story frame building was considered.
The wall covering of the bay was assumed to behave as a series of
closely-spaced beams which are simply supported between the adjacent
frames and act independently of one another~ It was further assumed
that the motion of the frames had no effect on the load transmitted
to them~ The pressure loading on the walls was considered to be a
triangular pulse with an initial peak. This conventionalized pulse
has been found to be a good approximation of the actual pressure
pulse.
Both elastic and plastiC analy-ses were made. In the plastic
analysis) it was assumed that a plastic. hinge forms at the center of
the beam. From these two analyses expressions for the dynamic end
reactions of the walls on the frames were derived, and charts were pre-
pared giving the variation of the dynamic reactions as a function of
time for different beam characteristics and different durations of the
load pulse~
This study is d.escribed in a doctoral disserta ti,on by W. J ..
Francy (Item 13 of Table I) and in a technical report by Francy and
Newmark (Item 21 of Table II).
(d) Influence of Foundation Coupling on the pynamic Response of ~imple otructures
This study was concerned with the deter.mj~tion of the effect
on the dynamic response of a structure of the interaction between the
structure and the foundation. The structure considered was a rectangular
64
box-shaped frame. The following two conditions of motion were inves
tigated: \1) the frame moves only as a rigid body, and (2) the frame
distorts as a shear beam. The exciting force was assumed to be a
pulse of rectangular shape. The quantities necessary to define this
force are the point of application of the force, its magnitude, and
its duration.
The flexibilities of the soil in the vertical and horizontal
directions were represented by the action of two vertical springs and
a horizontal spring. The springs were assumed to be linearly elastic
and weightless ..
For various combinations of the parameters considered, numeri
cal values of the responses of the structure were determined, from which
conclusions were deduced regarding the effect of interaction between
foundation and structure.
This investigation is described in the doctoral dissertation
by J. G. Hammer (Item 15 of Table I).
(e) ~fect of Vertical Loads on the nvnamic Response of structures
This study was concerned with the effect of vertical loads on
the ~namic response of a simple structure. The vertical load may be
the dead weight of the structure or may be produced by the vertical com
ponent of blast forces acting on the roo.f of the structure. Vertical
load decreases the resistance of the frame to horizontal loads and also
causes the initiation of yielding in the columns at a reduced value of
the horizontal forces.
In this investigation a method was developed for the analysis
of simple portal frames subjected to both vertical and horizontal load
ing. The procedure considers the behavior of the structure before and
after yielding and up to complete collapse. The load-deformation curve
neglecting vertical load is assumed to consist of a linear elastic seg
ment and a linear plastic segment having a slope different from that of
the elastic segment. This investigation included also a study of the
validity of some of the approximations used by Newmark in his design
procedure fer structures exposed to. atemic blast.
This study is described in a docteral dissertation by H. E.
Stephens (Item 26 of Table I).
CHAPrER V
MISCELLANEOUS
66
In this chapter are presented descriptions of the following
three unrelated investigations: (1) bounds and convergence of relaxa
tion procedures; (2) a method of numerical integration for transient
problems of heat conduction; and (3) two studies on the use of high
speed digital computing machines~
22. Bounds and Convergence of Relaxation and Iteration Procedures
In connection with the solution of the simultaneous, linear,
algebraic equations which arise in studies of stress analysi~ consider
able e~fort was expended in the projects reported herein on methods of
solution of the equations and, particularly, on the bounds and conver
gence of relaxation and iteration procedures. Some of the ideas devel
oped relative to these problems are summarized in a paper by N. M.
Newmark (Item 6 of Table III). This paper was reprinted and issued as
one of three reprints in a project technical report (Item 12 of Table
II).
To provide an indication of what is contained in this paper,
the abstract is quoted below:
"This paper discusses a number of general concepts relating
to relaxation procedures, methods of nsteepest descent," and iteration
procedures. The relationship between these is indicated for certain
systematic ways in which relaxation patterns may be developed. An
extrapolation procedure is developed for problems in which the successive
approximations technique, in its usual form, diverges. The procedure is
a generalization of an observation by Hartree and others that in many
cases such divergent problems can be treated by considering the diver-
gence to be a geometric one. The study of upper and lower bounds to
the errors in relaxation procedures is a second major part of the paper.
General procedures are developed for determining the maximum and mini-
mum errors in any of the quantities for which bounds to an influence
function can be determined. Use of the theorems derived makes it pos~
sible, for example, to estimate the error due to all sources including
IIround offn at any stage in the numerical solution of Laplace is or
Poisson's equation."
23. A Method of Numerical Integration for Transient Problems of Heat Conduction
In this study, a numerical method o.r integration was developed
for studying transient heat flow in solids. The canonical form of tbe
governing differential equation is o.f parabolic type, for which various
methods of numerical integration have been studied. The method devel-
oped in connection with this investigation is more general and versa-
tile than other existing methods. The two methods which give stable
solutions, studied by HYman, Kaplan, and Brien, are found to be two
special cases of the present method.
In this procedure, the differential operator with respect to
the spatial coordinates is replaced by its eigenvalues. This simplifies
68
the development, and the results can be interpreted in terms of eigen
modes of the differential operator, which are fOlUld to exist for a
great many physical problems, even with time-dependent conditions. A
spectrum for the truncational error is obtained with respect to the
eigenvalues of the system.
The procedure is essentially a method of step-by-step integra
tion. The time coordinate is first divided into a number of short inter
vals. For each step the temperature and its first time derivative at
the end of the step are calculated from the corresponding quantities at
the beginning of the step and the governing equation. By repeating this
process successively for all intervals of time within the range of inter
est, a complete solution is obtained. The time interval chosen for this
method must lie within certain limits to give stable solutions, and these
bounds were derived in this work.
Within each step the calculation of the temperature at the end
of the interval may be done either by the straightforward solution of
simultaneous equations or by an iterative procedure. The criterion for
convergence of the iterative procedure was derived.
The differential equation which governs the flow of heat in
solids is of the same type as that defining several other physical phe
nomena. For instance, the equation for consolidation of clay, the kinetic
equation of rate process, the diffusion equation in chemical reactions,
and the Fokker-Planck Equation in the theory of probability, are all of
the parabolic type treated in this note.
This work is described in a technical report by Tung and Newmark
(Item 22 of Table II).
24. studies on the Use of High Speed Digital Computing Machines
In' addition to the studies of the use of the digital computor
which were m~de in connection with the applications reported in other
sections herein, the two investigations described below were concerned
solely with the development of efficient computor programs for problems
of common occurre.nce.
A study of procedures for the solution of large numbers of
simultaneous linear eCluations was made by A. R. Robinson. The main
characteristics of different methods were compared. A new type of
iterative procedure was developed. This method has the ad\~ntage that
it is applicable to any system of eCluations having a uniClue solution.
However, the new procedure shares with all other known iterative methods
the disadvantage of converging extremely slowly for very badly condi
tioned eCluations. For such sets of eCluations, which are by no means
rare in applications, it seems that the Gauss elimination procedure, if
properly used, is the best computational scheme for obtaining results
in a reasonable amount of time. Yet the storage reCluirements of the
Gauss method seriously limit the number of eCluations which may be
solved on present-day digital computors.
This study is reported in a Ph.D. dissertation (Item 21 of
Table I).
A study of procedures for the solution of structural mechanics
problems on high speed digital computing machines was made by J. A.
Brooks. The primary object of the investigation was the development of
computor programs for the solution of complex structural dynamics prob
lems such as those associated with effects of explosions, vehicle impact,
and earthCluakes. As part of this general investigation the problems of
determining static deflections, buckling loads, natural freCluencies, and
70
transient response have been treated separately. The whole investiga
tion of problems of this t:r.pe may be subdivided into three phases
according to the type of work involved.
The first part of this work is the derivation of equations
of motion for the structure under study, where these equations are
required to fit a form for which methods 01 solution are available.
Preferably) the methods would already exist in coded form.
The second part of this work is the preparation of computor
programs for the resisting forces of the structure) the external
forces) and the residual forces. The residual force routine for the
equations of equilibrium is merely a combination of the resisting
force routine and the external force routine. The residual force
routine that enters into computations of the matrices in the buckling
problem is the same as that for the static deflection problem.
The third part of the work is the construction of the equa
tion solving codes and library routines. Ideally, this phase of the
work should be completed before the first two parts are undertaken.
The study is reported in a Ph.D. thesis (Item 5 of Table I).
In this thesis is outlined an approach to the solution of structural
mechanics problems in which the only coding work required .for study of
an individual structure is the preparation of a resisting force routine
and an external force routine. The remainder of the codes can be pre
pared as library routines which can be used for other structural prob
lems. In support of this approach) programs have been prepared for
71
finding the static deflections, buckling loads, and transient response
of an elastic arch which was used as an example.
As part of this study a numerical procedure for the solution
of the following determinental equation,
= 0
was prepared. The matrices A, B, C, D, .. are symmetrical and
real. The quantity A is a scalar representing the magnitude of the
load on the structure. The solution o.f the above equation gives the
buckling loads and buckling modes. The method of solution is based
on L9.nczos t method of minimized iterations. Also a program for the
solution of slightly non-linear simultaneous algebraic equations was
prepared. The method is based on the method of conjugate gradients.
1.
2.
3·
4.
5·
6.
7·
8.
9·
10.
11.
12.
13·
14.
72
TABLE I
LIST OF THESES PREPARED UNDER SPONSORSHIP OF CONTPACT N6cri-71, TASK ORDER VI
Aisawa, S.
Au, T.
Austin, W. J.
Badir, M.
Brooks, J. A.
Brooks, N. B.
Chan, S. p.
Chen, T. y.
Colin, E. C~
D!Appolonia, E.
Dauphin, E. L.
Dub erg , J. E.
Francy, W. J.
Gossard, M. L.
tiNumerical Procedures for One-Dimensional Vibration Problems," M.S. 1948
"A Numerical Procedure for the Analysis of Pressure Vessel Heads," Ph.D. 1951
alA Numerical Method for the Solution of Plate Buckling Proble..TUs, n Ph.D. 1949
"Torsional-Flexural Buckling of Beams and Columns by Approximate Methods of Analysis, U Ph. D .. 1948
"Solution of structural Mechanics Problems on High Speed Digital Computing Mg.chines," Ph.D. 1955
"The Response of Simple structures to D,ynamic Loads," M.S. 195.3
wComparison of Step-by-Step Methods for Analyzing the Dynamic Response of structures," Ph.D. 1953
~8Numerical. Determination of Influence Coefficients for Moments in Skew I-Beam Bridges/I Ph.D. 1954
nA Flow Analogy for Torsion, r~ M.S. 1947
taSolutions of Radially Symnetric stress Problems by Numerical Methods/! Ph.D. 1948
. uFramework Analogies for Plane stress Problems in ElastiCity," M.S. 1947
~A Numerical Procedure for the stress Analysis of Stiffened Shell Structures,9t Ph.D. 1948
itA study of Blast Loading Transmitted to Building Frames, Rt Ph.D. 1954
HAn Iterative Transformation Procedure for Solving Characteristic-Value Problems in Structural Mechanics," Ph.D. 1949
15· Hammer, J. G.
200 Pan, S. L.
21. Robinson, A. Ro
22. Rosenblueth, E.
23· Schnobrich, W. c.
24. Schmidt, R.
25· Schutz, F. w.
26. Stephens, H. E.
27· Sutherland, J. G.
280 Tung, C. P. C.
29· Tungj T. p.
73
TABLE I (CONTINUED)
liThe Influence of Foundation Coupling on the D,ynamic Response of Simple Structures," Ph.D. 1954
"A Numerical Procedure for Analyzing D,ynamic Structural Response," M.S. 1956
liA Numerical Procedure for Beams on ElasticSolid Subgrades," Ph.D. 1951
"Compatibility Procedures for Plane stress Problems, tt M. S . 1949
UA Numerical Procedure for the Solution of structural D,ynamics Problems with Concentrated Masses," Ph.D. 1950
"Influence of Ductility on the Response of Simple structures to Earthquake Motions," Ph.D. 1951
UAn Iterative Method for the Solution of a Large Number of Simultaneous Equations," Ph.D. 1955
itA Basis of Aseismic Design o-r Structures," Ph.D. 1951
"A Numerical Method for the Analysis of Cylindrical Shells,n MoS. 1955
aJApproximate Analysis of D::>mes, n Ph.D. 1956
"An Iteration Procedure for Bars on Elastic Foundations,~ M.S. 1950
"A Study of the Effect of Vertical Loads on the DynamiC Response of structures, Ii Ph.D. 1954
ttAnalysis of Plates Continuous Over Flexible Beams," Ph.D. 1953
"A Numerical Solution for Lateral and Torsional Buckling of Beams and Columns," Ph.D. 1951
"A Numerical Approach to Problem of Generalized Plane Stress," Ph.D. 1950
74
TABLE I (CONTINUED)
30. Veletsos, A. S. ftFlexural Buckling of Non-Prismatic, Elastically Restrained Bars," M.S. 1950
31. Veletsos, A. S. "A Method for Calculating the Natural Frequencies of Continuous Beams J Frames, and Certain Types of Plates,rt Ph. D. 1953
32. Wu, G. S. itA Numerical Procedure for the Analysis of Hipped-Plate structures," Ph.D. 1954
33· Wu, T. s. uEffect of Small Initial Irregularities on the strength of Cylindrical Shells,1t Ph.D. 1952
34. Yegian, S. "Lateral Buckling of I-Beams Supported by Cables," Ph.D. 1956
35· Yuan, H. K. "Numerical Studies of the Curved Beam Problem," Ph.D. 1951
TABLE II
LIsr OF TECHNICAL REPORTS ISSUED UNDER CONTRACT N6or i-7l, TASK ORDER VI
The following listing is arranged chronologically.
75
1. tiA Numerical Procedure for the Solution of Plate Buckling Problems, n
by Wo J. Austin and N~ M. Newmark, SRS 4)* August 1950.
2. "A Numerical Procedure for the Analysis of Pressure Vessel Heads," by T. Au, L. E. Goodlnan, and N. M. Newmark, SRS 10, February 1951.
3. "Influence of Ductility on the Response of Simple structures to Earthquake f.btions, n by S. L. Pan, L. E. Goodlnan, and N. M. Newmark, SRS 11, March 1951.
4. "Vibrations of Prismatic Bars Including Rotatory Inertia and Shear Corrections," by J. Go Sutherland and L. E. Goodman, SRS l2, April 1951.
5. "A Simple Approximation for the Natural Frequencies of Partly Restrained Bars," by N. M. Newmark and A. S. Veletsos, SRS 25, April 1952.
6. "Aseismic Design of Elastic Structures Founded on Firm Ground, n by L. E. Goodman, E. Rosenblueth, and No M. Newmark, SRS 26, June 1952.
7. uA Comparison of Numerical Methods for Analyzing the DynamiC Response of structures, n by N. M. Newmark and S. p. Chan, SRS 36, October 1952.
80 "Analysis and Design of Structures Subjected to Dynamic Loading, n
by N. M. Newmark, SRS 37, November 1952.
9· uA Method for Calculating the Natural Frequencies of Continuous Beams, tI by A. S. Veletsos and N. M. Newmark, SRS 38, January 1953 ..
10. "Computation of Dynamic Structural Response in the Range Approaching Failure,'· by N. M. Newmark, Reprint, SRS 41, December 1952.
* The notation SRS refers to the Structural Research Series of reports of the Civil Engineering Department of the University of lllinois.
TABLE II (CONTINUED)
li. "Analysis of Plates Continuous Over Flexible Beams," by J. G. Sutherland, L. E. Goodman, and N. M. Newmark, SRS 42, January 1953·
12. "Three Papers on Numerical Methods of Structural Analysis," by N. M. New'IIlark, E. ngAppolonia, and W. J. Austin, Reprints, SRS 47, 1952.
13. rtCircular-Crested Vibrations of An Elastic Solid Bounded by Two Parallel Planes,,! by L. E. Goodman, Reprint, SRS 48, 1952.
14. "Effect of Small Initial Irregularities on the Stresses in Cylindrical Sheils, t1l by T. S. Wu, L. E. Goodman, and N. M. Newmark, SRS 50, April 1953.
15. "The Response of Simple structures to D,ynamic Loads," by N. B. Brooks and N. M. Newmark, SRS 51, April 1953.
16. n A Method for Calculating the Natural Frequenc ies of Continuous Beams, Frames, and Certain Types of' Plates," by A. S. Veletsos and N. M. Newmark, SRS 58, June 1953.
17· "An Engineering Approach to Blast Resistant Design," by N. M. Newmark, Reprint j SRS 63, October 1953.
18. t1A Simple Approximation for the Fundamental Frequencies of TwoSpan and Three-Span Continuous Beams," by A. S. Veletsos and N. M. Newmark, SRS 66, February 1954.
190 "A Review of Numerical Integration Methods for D,ynamic Response of structures, n by ToP. Tung and N. M. Newmark, SRS 69, M3.rch 1954.
20. "Tables of Deflection and Moment Coefficients for the Steady-State Vibration of Uniform Bars/' by A. S. Veletsos and N. M. Newmark, SRS 71, M3.y 1954.
21. "A study of Blast Loading Transmitted to Building Frames," by W 0 J. Francy and No Mo Newmark" SRS 89, December 1954e
22. tlA Method of Numerical Integration for Transient Problems of Heat Conduction," by T. P. Tung and N. M. Newmark, SRS 95, March 1955.
23. tlLateral Buckl.ing of Elastically End-Restrained I-Beams," by W. J. Austin, S. Yegian, and To Po Tung, Reprint, SRS 101, April 1955.
77
TABLE II (CONTINUED)
24. "Two Papers on Natural Frequencies of Continuous Beams," by A. S. Veletsos and N. M. Newmark, Reprints, SRS 103, June 1955.
25. "Numerical Analysis of Earthquake Response of a Tall Building," by T. P. Tung and N. M. Newmark, Reprint, SRS llO, October 1955.
26. "A Statistical Estimate of Relative Distribution of Extreme Shear in a Tall Building Subjected to Random Earthquake Shocks," by T. P. Tung and N. M. Newmark, SRS 116, February 1956.
In addition to the reports listed above, the following are of the nature of technical reports, but were issued before the system of technical reports was begun.
27. "A Numerical Procedure for Solving Laplace's and Poisson's Equations, With Applications to Torsion and Other Problems," by E. C. Colin and N. M. Newmark, Progress Report No.1, November 1946.
28. "A Numerical Solution for the Torsion of Hollow Sections," by E. C. Colin and N. M. Newmark, Special Report, 18 March 1947.
TABLE III
LIST OF PUBLICATIONS PREPARED' UNDER SPONSORSHIP OF CONTRACT N6ori-71J TASK ORDER VI
78
1. "A Numerical Solution for the Torsion of Hollow Sections," by Eo C. Colin and N. M. Newmark, Journal of Applied Mechanics, January 1948.
2. "Numerical Methods of Analysis of Bars, Plates, and Elastic Bodies," by No M. Nevnnark, Chapter 9 of Numerical Methods of Analysis in Engineering, edited by L. E. Grinter, MacMillan, 1949.
3. "Analysis and Design of structures Subjected to Dynamic Loading, II by N. M. Newmark, Proceedings, .MIT Conference on Building in the Atomic Age, June 1952.
4. "Computation of Dynamic structural Response in the Range Approaching Failure" by N. Mo Newmark, Proceedings, UClA Symposium on Effects of Earth~uakes and Blast on structures, June 1952.
5. "A Numerical Method for the Solution of Plate Buckling Problems," by W. J. Austin and N. M. Newmark, Proceedings, First National Congress of Applied Mechanics, 1952~
6. "Bounds and Convergence of Relaxation and Iteration Procedures," by N. M. Newmark, Proceedings, First National Congress of Applied Mechanics, 1952.
7. "A Method for the Solution of the Restrained Cylinder Under Compression,~ by E. DtAppolonia and N. M. Newmark, Proceedings, First National Congress of Applied Mechanics, 1952.
8. "Circular-Crested Vibrations of an Elastic Solid Bounded by Two Parallel Planes, Ri by L. E. Goodman, Proceedings, First National Congress of Applied ~chanics, 19520
9. "An Exact Solution for a Rectangular Plate Problem," by J. Go Sutherland, Journal of Applied Mechanics, September 1952.
10. "A S:iJ:nple Approximation ,for the Natural Frequencies of Partly Restrained Bars," by N. M. Newmark and A. S. Veletsos, Journal of Applied Mechanics, December 1952.
11. "Aseismic Design of Fir.mly Founded Elastic Structures," by L. Eo Goodman, E. Rosenblueth, and No M. Newmark, Trans. ASCE, Vol. 120, 1955, po 782.
79
TABLE III (CONT INUED )
12. "Determination of the Natural Frequencies of Continuous Beams on Flexible Supports," by A. S. Veletsos and N. M. Newmark, Proceedings, Second U. S. National Congress of Applied Mechanics, 1954.
13. tlA Simple Approximation for the Fundamental Frequencies of Two-Span and Three-Span Continuous Beams, n by A. S. Veletsos and N. M. Newmark, Proceedings, Second U. S. National Congress of Applied Mechanics, 1954.
14. "Lateral Buckling of Elastically End-Restrained I-Beams," by W. J. Austin, S. Yegian, and T. p. Tung, ASCE Proceedings Separate No. 673, April 1955·
15. "Natural Frequencies of Continuous Flexural Members," by A. S. Veletsos and N. M. Newmark, ASCE Proceedings Paper No. 735, July 1955·
16. IIAn Engineering Approach to Blast Resistant Design," by N. M. Newmark, Transactions, ASCE, Vol. 121, 1956. Also available as University of Illinois Engineering Experiment Station Bulletin, Reprint Series No. 56.
17. "Determination of Natural Frequencies of Continuous Plates Hinged Along Two Opposite Edges," by A. S. Veletsos and N. M. Newmark, JOlJ~DB~l of Applied MecPRnics) March 1956=
18. "Numerical AnalYSis ofEB.rthquake Response of a Tall Building," by T. p. Tung and N. M. Newmark, Bulletin of the Seismological Society of America, Vol. 45, October 1955.
19. "Shears in a Tall Building Subjected to Strong Motion Earthquakes," by T. P. Tung and N. M. New~rk. Proceedings of the World Conference on Earthquake Engineering, June 1956, ppo 10-1 - 10-11.
20. Discussion by L. E. Goodman and J. G. Sutherland of a paper entitled "Natural Frequencies of Continuous Beams of Uniform Span Length," Journal of Applied Mechanics, Vol. 18, 1951, pp. 217-218.
21. Discussion by A. S. Veletsos of a paper entitled "Steady-State Forced Vibration of Continuous Frames, n Trans. ASCE, Vol. 118, 1953, pp. 817-820.
a
nrsrRIBUTIoN LIm - PROJreT NR-064-183 - TASK VI
Administrative Reference and Liaison Activities
Chief of Naval Research Department of the Navy Washington 25, D. C. ATTN: Code 438
Code 432 Code 423
Director Naval Research Laboratory Washington 25, D. C. ATTN: Tech. Info. Officer
Technical Library Mechanics Division
Commanding Officer Office of Naval Research Branch Office 495 Summer street
(4) (1) (1)
(6) (1) (2)
Boston la, Massachusetts (1)
Commanding Officer Office of Naval Research Branch Office 3 46 Broadway New York 13" New York (1)
Office of Naval Research The John Crerar Library Bldg. lOth Floor, 86 Eo Randolph st. Chicago 1, IlIDlois
Commander
(2)
U. S. Naval Ordnance Test Station Inyokern, China Lake, California ATTN: Code 501 (1)
Commander U. S. Naval Proving Grounds Dahlgren, Virginia (1)
Armed Services Tecb~ical Information Agency
Documents Service Center Knott Building Dayton 2, Ohio
Commanding Officer Office of Naval Research Branch Office 801 Donahue street San Francisco 24, California
Commanding Officer Office of Naval Research Branch Office 1030 Green street Pasadena, California
Officer in Charge Office of Naval Research Branch Office, London Navy No. 100 FPC, New York, New York
Exchange and Gift Div. Library of Congress Washington 25, D. C.
Commander
(1)
(1)
(1)
(2)
U. S. Naval Ordnance Test Station Pasadena Annex 3202 E. Foothill Blvd. Pasadena 8, California ATTN: Code p8087 (1)
Department of Defense Other Interested Government Activities GENERAL
Research and ~lopment Board Department of Defense Pentagon Building Washington 25, D. C. ATTN~ Library (Code 3D-I075) (1)
Armed Forces Special Weapons Project P. O. Box 2610 Washington, D. C. (1) ATTN: Chief, Weapons Effects Division
Chief, Blast Branch (2)
ARMY
Chief of Staff Department oi' the Army Research 2nd Development Div. Washington 25; D. C. ATTN: Chief of Rese~rch and
Development (1)
Office of the Chief of Engineers Assistant Chief for Public }lorks Department of the Army Bldg. T-7; Gravelly Point Washington 25, D. C. ATTN: Struc tural Branch
(R. L. Bloor) (1)
Chief of Engineers Engineering Division MilitarJ Construction Washington 25, D C. A'ITN: ENGEB ( 2 )
Engineering Resesrch and Development L3..boratory
Fort Belvoir;1 ViI'ginia ATTN: structures Branch
The Commanding Gener~l Sandia Base, P.O. Box 5100
(1)
A1bu<luerque, NevT Mexico (1 )
Corps of Engineers, U. S. Army Ohio River Division Libs 0
5851 Mariemont Ave., ~~riemont Cincinnati 27, Ohio ATTN: F. M. Mellinger (2)
Operations Research Officer The John Hopkins University 6410 Connecticut Avenue ChevJ Chase; M:1ryland
Office of Chief of Ordnance Research and Development Service Department of the Army The Pentagon Washington 25, D. C.
(1)
ATTN: ORDrB ( 2 )
Ballistic Research Laboratory Aberdeen Proving Ground Aberdeen, M3.ryland
b
ATTN: Dr. C. H. L3.mpson (1)
Commanding Officer watertown Arsenal \{a tertovm, M3.ssachusetts ATTN: Laboratory Division (1)
Commanding Officer Frankford Arsenal Philadelphia , Pennsylvania A'ITN~ Laboratory Division (1)
Commanding Officer Squier Signal Laboratory Fort Monmouth, Ne\v Jersey ATTN: Components and
Materials Branch (1)
Other Interested Government Activities
NAVY
Chief, Bureau of Ships Navy Department Washington 25: D. C. ATTN: Director of Research
Code 449 Code 430 Code 421 Code 423 Code 442
Director David Taylor MOdel Basin Washington 7, D. C. ATTN: structural Mechanics
Division
Director Naval Engineering Experiment
Station Anna pol is, Mlryland
(2) (1) (1) (1) (1) (1)
(2)
(1)
Director Materials Laboratory New York Naval Shipyard Brooklyn 1, Nei,{ York
Chief, Bureau of Ordnance Navy .Department Washington 25, D. C. ATTN: Ad-3, Technical Lib.
Rec ., T. N. Girauard
Super intendent Naval Gun Factory
(1)
(1) (1)
Washington 25, D. C. (1)
Naval Ordnance Laboratory White Oak, M3..ryland RFD 1) Sil ver Spr ing, M9.ry land ATTN: Mechanics Division (2)
Naval Ordnance Test Station Inyokern, China L:1ke, California ATTN: Scientific Officer (1)
Chief of Bureau of Aeronautics Navy Department Washington 25, D. C. ATTN: TD-41, Tech. Lib. (1)
DE-22, C. W. EQTley (1) DE-23, E. M. Ryan (1)
Super intendent Post Graduate School U. S. Naval Academy Monterey, California (1)
Naval Air Experimental Station Naval Air Materiel Center Naval Base Philadelphia 12, Pennsylvania ATTN: Head, Aeronautical
Materials Laboratory (1)
Chief, Bureau of Yards and Docks Navy Department Washington 25, D. C. ATTN: CodeP-3l4 (1)
Code C-313 (1)
c
Officer in Charge Naval Civil Engineering Research
and Evaluation Laboratory Naval Station Port Hueneme, California (1)
Cormnander U. S. Naval Ordnance Test Station Inyokern, China Lake, California (1)
AIR FORCES
Commanding General U. S. Air Forces The Pentagon Washington 25, D. C. ATTN: Research and Development
Division (1)
Chief, Structures Division Research Directorate Air Force Special Weapons Center Kirtland Air Force Base New Mexico ATrN: SWRS, Eric H. Wang (2)
Office of Air Research Wri&~t-Pdtterson Air Force Base Dayton, Ohio ATTN: Chief, Applied M2chanic s
Group (1)
Director of Intelligence Headquarters, U. S. Air Force Washington 25, D. C. ATrN: Air Targets Division
Physical Vulnerability Div. AFOIN-3B (2)
OTHER GOVERNMENT AGENC IES
U. S. Atomic Energy Commission Division of Research Washington, D. C. (1)
Argonne National Laboratory Bailey and Bluff Lamont, Illinois (1)
Director National BUTeau of Standards Washington, D. C. ATrN: Dr. W. H. Ramberg
U. S. Coast Guard 1300 E Street, NoW. Washington, D. C. ATrN: Chief, Testing and
Development Division
Forest Products Laboratory Madison, Wisconsin ATrN: L. J. Mir bvard t
National Advisory Committee for Aeronautics
1512 H street, N.W. Washington 25, D. c.
National Advisory Committee for Aeronautics
Langley Field, Virginia ATTN: Structures Lab.
D,ynamic Loads Branch
National Advisory Committee for Aeronautics
Cleveland MUnicipal Airport Cleveland, Ohio ATTN: J. H. Collins, Jr.
U. S. Maritime Commission Technical Bureau Washington, D. C. ATTN: Mr. V. Russo
(2)
(1)
(1)
(1)
(1) (1)
(1)
(1)
Contractors and Other Investigators Actively Engaged in Related Research
Professor lifnn Beedle Fritz Engineering Laboratory Lehigh University Bethlehem, Pennsylvania
Professor R. L. Bisplinghoff Massachusetts Institute of Technology Cambridge 39, ~ssachusetts (1)
Dr. Walter Bleakney Department of PhysiCS Princeton University
d
Princeton, Nevi Jersey (1)
Dean H. L. Bm-nna.n College of Engineering Drexel Institute of Technology Philadelphia, Pennsylvania (1)
Chairman, Dept. of Aeronautics The JohnB Hopkins University School of Engineering Baltimore 18, M:Lryland (1)
Professor T. J. Dolan Dept. o.f Theoretical and
Applied Mechanics University of Illinois Ul~bana, Illinois (2)
Professor Lloyd Donnell Department of Mechanics Illinois Institute of Technology Technology center Chicago 16, Illinois
Dr. D. C. Drucker
(1)
Chairman, Division of Engineering Brown University Providence, Rhode Island (1)
Professor W. J. Duncan, Head Dept. of Aeronautics' James w~tt Engineering Labs The University Glasgow W. 2 England
Dean W. L. Everitt College of Engineering University of Illinois Urbana, Illinois
fic. L. Fox Mathematics Division National Physical Laboratory Teddington, Middlesex England
(1)
(1)
(1)
Armour Research Foundation 3422 S. Dearborn Chicago 16, Illinois ATTN: Dr. G. Notbmann
Professor B· Fried Washington state College Pullman, Washington
(1)
(1)
Dr. M3.rtin Goland, Vice President Southwest Research Institute 8500 Culebra Road San Antonio, Texas (1)
Dr. J. N. Goodier School of Engineer.ing Stanford University Stanford, California (1)
Professor L. E. Goodman Dept. of Mechanics and Mlterials University of Mirmesota Mdnneapolis, Minnesota (1)
Professor A. E. Green Kings College Ne\>Tcastle on Tyne, 1, England
Dr. R. J. Hansen Massachusetts L~stitute of
Technology Cambridge 39, Massachusetts
Professor R. M. Hermes University of Santa Clara Santa Clara, California
Dr. N. J. Hoff, Head Department of Aeronautical
(1)
(1)
(1)
Engineering and Applied Mechanics Polytechnic Institute of Brooklyn 99 Livingston Street Brooklyn 2, Nevi York (1)
Dr. W. H. Hoppmann Dept of Applied Mechanics Johns Hopkins University Baltimore, M:1ry land (1)
Professor L. S. Jacobsen Stanford University Stanford, California
Dr. Bruce Johnston 301 W. Engineering Building University of Michigan Ann Arbor, Michigan
Professor W. K. Krefeld College of Engineering Columbia University New York, New York
Professor B. J. Lazan Department of MechaniCS University of Minnesota Minneapolis 14, :Minnesota
Department of Mechanics Rensselaer Polytechnic lnst. Troy, New York ATTN: Prof. Geo. Lee
Prof. W. R. Osgood
Dr. M. M. Lemcoe Southwest Research Institute 8500 Culebra Road
e
(1)
(1)
(1)
(1)
(1) (1)
San Antonio, Texas (1)
Library, Engineering Foundation 29 West 39th Street New York, New York (1)
Sandia Corporation Sandia Base Albuquerque, New Mexic 0
ATTN~ Dr. M. L. ~rritt (1)
Professor N. M. Newmark, Head Department of Civil Engineering University of Illinois Urbana, Illinois (2)
Professor Jesse Ormondroyd University of Michigan Ann Arbor, Michigan (1)
Dr. ~.{. Prager) ChJ.irmD.n Pb;)rs ico.l Sc iences Counc il Brovrn University Proi.ridence; Rhodc= Islu.nd
Rand Corpor~ticn Nuclear Energy Division 1700 ~~in street Santa Monica, C~lifornia ATTN: Mr. M;::.rc Peter, Jr.
Dr. R. LJ.tter
Dr . S. Raynor Mechanics Research Dept. American Machine and Foundry Co. 188 ~Y. Rano..olph street
(1)
(1) (1)
Chicago 1, Illinois (1)
Professor E. Reissner Department of V~thematics Massachusetts Institute of
Technology Cambridge 39, Massachusetts (1)
Professor F. E. Richart, Jr. Dept. of Civil Engineering University of Florida Gainesville, Florida
Dr. J. D. Shreve, Jr. Sandia Corporation ~ndia Base Albuquer<lue, Ne\[ Mexico
Dr. C. B. Smith Department of ~nthematics Wall-;:er Hall University of Florida
(1)
(1)
Gainesville, Florida (1)
Professor R. V. south~.vell
The Old House, Trumpington Cambridge, England
Professor E. Sternberg Division of Engineering BrOl~'ll Uni ver s i ty P .. e ovidenc e, Rhode Island (1)
f
Professor F. K. Teichmann Dept. of Aeronautical Engineering Ne\v York University University Heights, Bronx NeH York, Ne,{ York (1)
Dr. G. E. Uhlenbeck Engineering Research Institute University of Michigan .Ann Arbor, Michigan (1 )
Dean F. T. Wall Graduate College University of lllinois Urbana, Illinois (1)
Dept. of Aeronautical Engineering NevT York Universit;:/ University Heights, Bronx Nel:T York) Ne-vl York (1)
Dr. M. P. ~fuite
Department of Civil Engineering University of ~nssachusetts Awnerst, l~ssachusetts (1)
TASK VI PROJ:Er!T - C. E. RESEARCH STAFF
Dr. W. J. Austin (1)
Dr. T. y. Chen (1)
Dr. A. S. Veletsos (1)
Professor W. H. Mlnse (1)
Research Assistants (5)
Files (5)
Feserve (20)
Recommended